My current main focus is in solving the simplest example of an interacting quantum field theory in four dimensions. The example in mind is N=4 SYM which is a non trivial conformal theory with maximal supersymmetry. Solving the theory means to be able to efficiently compute any observable in the theory at any values of the parameters. Solving an interacting QFT in four dimensions would be a huge theoretical achievement. I believe it will have implications on our understanding of nature. It will also become part of the text books of tomorrow and will play an analog role to the harmonic oscillator in quantum mechanics.
N=4 SYM is believed to be integrable (at least in the planar limit) and integrability is the main tool we use to solve it. So far, the problem of computing anomalous dimensions of all local operators in the theory has been solved. We are now mainly working on computing scattering amplitudes (or equivalently, polygon Wilson loops) using integrability. We have also started computing correlation functions, which is the next step in complexity.