Dirac once said that renormalization is just a stop gap procedure, and there had to occur a fundamental change in our ideas. Did something change? Once, Dirac said the following about renormalization in Quantum Field Theory (look here, for example):

Renormalization is just a stop-gap procedure. There must be some fundamental change in our ideas, probably a change just as fundamental as the passage from Bohr's orbit theory to quantum mechanics. When you get a number turning out to be infinite which ought to be finite, you should admit that there is something wrong with your equations, and not hope that you can get a good theory just by doctoring up that number.

Has this fundamental change come along afterward, and if so, what is the nature of this "fundamental" change? Is it an attempt to unify quantum mechanics with general relativity (of which the two main streams are String Theory and Loop Quantum Gravity, and of which I don't think they correspond with reality, but that aside)? Is there something more exotic? Or was Dirac just wrong by assuming that the procedure is just a "stop-gap" procedure?
 A: Why do you think something has changed? Although I totally agree with ACuriousMind's viewpoint that viewing QFT as an effective theory relieves some of the pressure about the nature of renormalization, I don't think that is what Dirac was envisioning.
As mentioned in the comments, Dirac was a geometer at heart - or at least, one who appreciated the mathematical structure of the universe. In that context, I don't think he would have viewed the shift to effective field theory as very attractive. I think he meant "fundamental" as including a fundamental shift in our understanding of the mathematical structure of our universe.
So in my view, he would have viewed attempts to modify that structure (via strings, loops, non-commutation, etc) as the fundamental change needed. So from that perspective, there has not been the change that Dirac was seeking. We are still working on that.
A: Regarding the question of whether such fundamental change has come along, I would say yes and no. Yes because Wilson's point of view has provided a much clearer picture of renormalization which by the way is not "sweeping divergences under a rug". Dirac's statement about his understanding of renormalization (or lack thereof) is quite obsolete in 2017. However, I also said no because the development of Wilson's RG is in my opinion still work in progress. I know that this by now is textbook material for QFT courses, but I think Wilson's RG is not yet understood in the more general situation where couplings (and cut-offs!) are space-dependent. My feeling is that ultimately one can only declare victory and say "yes we now understand Wilson's RG" when we understand its relation with the holographic RG and perhaps the AdS/CFT correspondence itself. 

Edit: For those who still think that renormalization is a "stop gap procedure" or some kind arbitrary cookbook recipe, see my answer to Wilsonian definition of renormalizability . It should (hopefully) make it clear that renormalization of continuum QFTs viewed within Wilson's framework is in fact a well-posed and, I dare say, beautiful mathematical question. Some, like perhaps Dirac, might entertain the idea that BPHZ or Wilson-Polchinski renormalization is a patch awaiting a better more conceptual or geometric explanation. My answer is not at all orthogonal to this belief. This is what I tried to point out when mentioning the connection to the holographic RG which is a kind of geometrization  of the RG via the introduction of an extra coordinate direction $z$ corresponding to scale. Experts of AdS/CFT can correct me if I am wrong, but my understanding is that the connection to the RG is believed to play an important role in this correspondence, but the precise quantitative connection between Wilsonian RG and holographic RG remains elusive.
A: I think that Dirac was unsatisfied with the mathematical meaninglessness of the way renormalization was done. 
This has changed with causal perturbation theory. The latter is a covariant and mathematically impeccable way of handling perturbative UV renormalization, without introducing anywhere during the development a cutoff (and the associated dubious limit), a geometrically meaningless nonintegral dimension, or any mathematically undefined (infinite) quantity. Neither are there any unphysical bare quantities - the parameters $e$ and $m$ that occur in the causal approach to QED have throughout their physical meaning of the electron charge (at zero energy) and the electron mass. 
The problem with an energy cutoff $\Lambda$is that it destroys covariance and the causal structure, which appear only in the limit. Moreover, it leads (except in asymptotically free theories) to artifacts such as Landau poles, which prohibit taking the limit $\Lambda\to\infty$. A covariant approach that accounts for causality from the beginning avoids the latter and is conceptually superior. It also explains why the standard approach leads to the conventional problems with the infinities - namely because distributions can be multiplied only under carefully controlled conditions.
(The causal approach is perturbative but supports a renormalization group, which adds to it the same nonperturbative information as any RG enhanced perturbation theory, including an estimate for a possible Landau pole. The Landau pole is constructively dangerous only in an approach where a cutoff must pass the Landau pole. Thus even though there might be a Landau pole in the Bogoliubov-Stueckelberg renormalization structure present in the causal approach, it has no consequences at all, since one can do the perturbative construction at any fixed energy below the renormalization scale (in QED, even at $E=0$) and has a valid perturbation theory. Only the physical coupling needs to be small.) 
I don't think Dirac would have asked for a paradigm shift if he had known the causal approach and that it applies universally to all relativistic QFTs including the standard model. I believe that Dirac would have been satisfied with this resolution of his concerns. In any case, it completely removes his complaint 

When you get a number turning out to be infinite which ought to be
  finite, you should admit that there is something wrong with your
  equations, and not hope that you can get a good theory just by
  doctoring up that number.

The same holds for Feynman's complaint (cited in http://www.cgoakley.org/qft/ )

The shell game that we play ... is technically called
  'renormalization'. But no matter how clever the word, it is still what
  I would call a dippy process! Having to resort to such hocus-pocus has
  prevented us from proving that the theory of quantum electrodynamics
  is mathematically self-consistent. It's surprising that the theory
  still hasn't been proved self-consistent one way or the other by now;
  I suspect that renormalization is not mathematically legitimate.

With causal perturbation theory, perturbative renormalization has become mathematically fully legitimate and is well-understood.
A: There are a lot of projects going on, and I'll try to sum them up with pithy one-liners that are as accurate as my own (admittedly limited) understanding of them. The solutions include:


*

*Classical renormalization: it's the predictions that matter, and renormalization is just the only (admittedly complicated) way of taking the continuum limit we have.

*Wilsonian renormalization: it's simply not possible to construct a non-trivial theory that is not a low energy effective theory, and the non-renormalizable constants are those that don't affect low energy effective theories.

*String theory: this whole 4-d space-time is an illusion that is built from the interaction of interacting 2-d space-times (strings). Because all interactions are renormalizable in 2-d, the problems go away (though there are many compactified  space-like dimensions that we have yet to see).

*Loop quantum gravity: the problem comes from taking the continuum limit in space-time, so let's just throw out the idea of a continuum altogether.


I don't find any of these approaches particularly satisfying. My own inclination is to favor the "more derivatives" approach because it involves the fewest technical changes, but it requires an enormous philosophical change. The cause of that philosophical change comes about from the requirement that the theory be Lorentz invariant; it would, in principle, be possible to make theories not just renormalizable, but UV finite, by adding some more spatial derivatives. Because of Lorentz invariance, though, adding more space derivatives necessarily entails adding more time derivatives. Ostrogradsky showed in classical physics alone that more than two derivatives necessarily entails the Hamiltonian no longer having a lower bound (a good technical overview is given in Woodard (2007) and Woodard (2015)).
It is generally considered so important that the Hamiltonian serves as the thing that constrains the theory to a finite volume of phase space that it is half of one of the axioms that goes in to QFT; in sum:


*

*there exists an operator that corresponds to the Hamiltonian that serves as the generator of time translations (and to the Noether charge conserved due to the time invariance of the laws of physics), and

*the eigenvalues of the generator of time translations are positive semi-definite (or, have a lower bound).


The content of the Källen—Lehmann representation (Wikipedia link, also covered in section 10.7 of Weinberg's "The Quantum Theory of Fields", Vol. I) is that the above postulate, combined with Lorentz invariance, necessarily implies no more than two derivatives in the inverse of the propagator. 
The combination of Ostrogradsky and Källen—Lehmann seems insuperable, but only if you're insistent on maintaining that "Hamiltonian = energy" (here, I use "Hamiltonian" as shorthand for the generator of time translations, and "energy" as shorthand for "that conserved charge that has a lower bound and confines the fields in phase space"). I suspect that if you're willing to split those two jobs up that the difficulties in higher derivative theories disappear. The new version of the energy/time translation postulate would be something like:


*

*the generators of space-time translations are conserved (Hamiltonian, 4-momentum),

*there exists a conserved 4-vector operator that takes on values in the forward light cone, and

*The operators in 1 and 2 coincide for low frequency (classical physics correspondence).


A key paper in this direction is Kaparulin, Lyakhovich, and Sharapov (2014) "Classical and quantum stability of higher-derivative dynamics" (and the papers that cite it, especially by the same authors), which shows that the instability only becomes a problem for the Pais—Uhlenbeck oscillator when you couple the higher derivative sector to other sectors in certain ways, and it's stable when you limit the couplings to other ways.
All of that said, more derivatives wouldn't be a panacea. If you try to remove the divergences in a gauge theory by adding more derivatives, for instance, you'll always add interaction terms with more derivatives in such a way as to keep the theory as divergent as it was in the beginning. Note, that "more derivatives" is mathematically equivalent to Pauli—Villars regularization (PV) by partial fraction decomposition of the Fourier transform of the propagator. PV is known to not play well with gauge theory precisely because of this issue, although it's usually worded as violating gauge invariance because the higher order couplings with more derivatives required to keep gauge invariance are left out.
A: As Heterotic said in the comments, the "fundamental" change, regardless of how fundamental you think it really is, is most likely the change from the old view of renormalization as an arbitary choice of constants to hide unpleasant divergent quantities to the modern Wilsonian notion of the renormalization (semi-)group where the renormalization scale inherently represents a cutoff up to where the QFT considered is valid as an effective field theory - see also this answer of mine for an example of how the two views differ in viewing the renormalization scale. 
The fundamental change could therefore be flippantly phrased as the change from viewing QFTs as a fundamental theory of everything to using them as effective field theories with an inherent constraint on validity given by the Wilsonian cutoff.
A: He isn't the only one. Feynman also called renormalization' hocus-pocus'.
It turns out that renormalization has a nice mathematical structure described by the cosmic Galois group:

What is called the cosmic Galois group is a motivic Galois group that naturally acts on structures in renormalization in quantum field theory. The actual renormalization group is a 1-parameter subgroup of the cosmic Galois group.

It's heartening to see that a renormalization group is actually a group in the mathematical sense.
Also, renormalization is thought to be required to avoid infinities arising from the point particle idealization in QFT; and it turns out that renormalization itself is dispensed within perturbative string theory where point particles are replaced with strings. 
A: I am sure that the breakthrough should come from mathematics. Particularly, we will become able to manipulate the values of divergent integrals and series just like we manipulate real and complex numbers. These extended numbers would be as instrumental in QFT and vacuum physics as complex numbers in quantum mechanics.
