What are the main problems remaining to create a Theory of Everything (TOE) and who are some of the main contemporaries pursuing the TOE?
The current situation of physics is that we have a standard model of particle physics, which (if we ignore astronomical phenomena) accounts for most things, and which may easily (but a little arbitrarily) be extended to include neutrino masses, etc. So that's already a "unified theory", in the sense that it's a single theory that explains most phenomena.
That synthesis has been around since the 1970s. Since then, physicists have learned how to construct "grand unified theories" (in which the forces except gravity are united within one primordial force), supersymmetric theories (in which matter and force particles come in pairs), and finally string theory, which naturally subsumes all the preceding ideas and also the older idea of extra dimensions.
There has been enormous conceptual and mathematical progress in this theoretical realm, with significant contributions from hundreds of people. In particular, it's very natural to think of string theory as an ultimate context for all these possible theories. Choose a specific shape for the extra dimensions, and specify the other details, and string theory will reduce to a particular field theory, with all the parameters of the field theory (coupling constants, etc) being determined by the stringy geometry. String theory not only gives us a deeper source for the symmetry groups and particle representations appearing in the unified field theories, it also determines the parameters whose values are completely unexplained at the level of field theory.
In the contemporary world of theoretical physics, string theory has become an ultimate framework of possible theories, and you could divide theoretical labor into those who work more on the mathematical fundamentals, and then there are those who are crafting the rather intricate models needed to match reality, the phenomenologists. There are still people who work purely at the level of field theory, but if you think strings are needed to incorporate gravity, then that work has its greatest significance when placed in the context of string theory.
When the Large Hadron Collider was turned on, two things were supposed to happen: the Higgs boson would be found, and other particles were supposed to show up as well. Something like the Higgs was implied by the standard explanation for the W and Z bosons having mass, and something like supersymmetry was implied for why the effective mass of the Higgs can remain as light as it is. Those extra particles were also going to help the phenomenologists figure out what needs to be added to the standard model.
Extra particles could still turn up, but for now we have an austere and slightly puzzling situation, in which only the Higgs has appeared. At the same time, proton decay (a common side effect of grand unification) has not been detected, nor have dark matter particles (which supersymmetry could also have explained, if they were there).
We have a situation that in a way is the reverse of the 1960s, the decade immediately before the standard model. In the 1960s there were hundreds of facts about particle physics but no theory that contained them all. Now we have a single theory, the standard model, which may easily but arbitrarily be extended to explain things like neutrino masses, and we have an enormous theoretical edifice, the landscape of field theories that may be obtained from string theory.
So one possible attitude is to say that the next big step will consist of locating our universe in that landscape of possible theories, by making inspired use of the clues that we have. There are numerical hints among the parameters of the standard model, including some that have been known since the beginning of grand unification. There are reasons to doubt the standard theories of dark matter. And maybe when you put it all together, it becomes obvious that our universe is a double E-string compactified on the unit octonions (that's what I was thinking about this morning), or some such thing.
But no such breakthrough has happened yet. Instead we still have a babel of hundreds or even thousands of researchers, working within the unified string/field paradigm, pushing in different directions, and then just as many who are outside it, partly or entirely. And for now we just don't know who, if anyone, will take us to the next level.