Unitarity and renormalizability What is the difference between the unitarity of the theory and its renormalizability? Can we say that renormalizable theory is unitary after renormalization? 
The questions have arisen after I have seen that in some book (I forgot in which one) there were elements of proof of the unitarity of gauge theory after completing the proof of its renormalizability.  
 A: To put it simply:
Renormalizability is the feature that the theory you know at low energy scales can be extrapolated to "arbitrarily high" energy scales, without losing consistency.
Now some observations:


*

*When you say that a theory comes with a cutoff and seemingly doesn't work beyond that cutoff, then you're seeing something (that is scale dependent) break down, eg: unitarity. I don't know examples of anything else breaking down (I don't think it's feasible for gauge invariance or Lorentz invariance to break down at some scale, provided your gauge theory is free of anomalies)
Have a look at @JeffDror's answer and the comments on that.
Generally, when you see a non-renormalizable theory that breaks down at some energy scale, it is possible to improve its high-energy behaviour by adding more degrees of freedom. (Is this always possible? I don't have a clean argument, or any intuition for this.)

*Conversely, if a theory is renormalizable, then it better be free of unitarity violation -- at least when you extrapolate it into the UV. But extrapolating it into the IR might be quite a different matter.

*If you have a non-unitary theory, that means that you have some kind of dissipation. In such a case, unless you can protect yourself from dissipating energy/probability into those "hidden" modes, I don't think you'll have a pleasant time trying to extrapolate it into the UV. Operators which are irrelevant in the UV become relevant in the IR.
I think that unitarity is a "fundamental" feature lying at the heart of quantum mechanics (as we know it).  Lack of unitarity in a quantum theory signals serious loss of predictability or some manner of bad behaviour. On the other hand, any non-renormalizability seems to be a consequence of your over-enthusiasm in extrapolating your theory to higher energy scales -- which often results in something important going wrong -- and there aren't too many other things that can seemingly go wrong apart from unitarity.

This answer expresses my opinions, so it's not authoritative. I appreciate more comments and any corrections.
A: Unitarity says that the probabilities of any event is less then $1$. This is obviously an essential requirement for a given quantum theory and if a theory is not unitary then for it to describe Nature, it is necessarily missing some information that will fix this issue (such as new states and/or interactions).
Renormalizability just says that the theory requires a finite number of counterterms to make predictions. Once this set of finite terms is calculated, you can go on and make predictions for scattering events and forget that you ever needed to renormalize your theory. This is quite a special property and is by no means an essential requirement for a theory and one can develop quantum theories that don't have this property. 
The fact that renormalizability and unitarity are ever related to one another is a subtle property of quantum field theories. The reason is that a theory is nonrenormalizable if it has couplings constants of negative mass dimension (I don't know of a simple reason but I'm sure there is one...). These negative mass dimension terms can lead to nonunitarity in a theory at high energies.
To see how this works consider a real scalar field theory with a nonrenormalizable operator,
\begin{equation}
\Delta {\cal L} =\frac{1}{6!}  \frac{\lambda}{M^2} \phi^6
\end{equation}
If you now consider the scattering of 2 $\phi$ to produce 4 $\phi$ the amplitude is given by,
\begin{equation}
{\cal M} = \frac{\lambda}{M^2} \Rightarrow \left|{\cal M}\right|^2 = \frac{\lambda^2}{M^4} 
\end{equation}
But cross sections have units of $mass^{-2}$ and so by dimensional analysis the cross section must be roughly of the form,
\begin{equation}
\sigma \sim \frac{\lambda^2}{M^4} p^2
\end{equation}
where $p$ is the momentum of the process (this may look a bit different for a massive theory but lets take it to be massless for simplicity). 
Notice that the cross section blows up as $p$ gets very large. Therefore at one point the probability will get larger then $1$. Therefore, this theory must not be unitary. Since we know that nonrenormalizable theories aren't unitary we think of these theories as effective theories which require new interactions at around the scale that they break unitarity[*].
Lastly note that to the fix the unitarity problem we need momentum dependence in the denomenator. This comes from a propagator instead of a coupling and hence new states. 
[*]This is the general trend I don't know if this is a rigorous statement or not.
