Groups play a central role in just about all the branches of mathematics and continue to be a very active area of research as evidenced by the recent Field Medals awarded in the area. At present we have the culmination of a three directional attack on the Burnside problems. The second is the positive solution of the restricted Burnside Problem for residually finite groups by Zelmanov, and the third is the p-adic analytic methods in dealing with questions of linearity of residually finite groups by Alex Lubotzky and Avinoam Mann. There are also the remarkable advances made by Aner Shalev, Lubotzky, and others on pro-finite groups and results of Dan Segal and others for residually finite solvable groups.
Representation theory continues of fundamental importance in mathematics and other sciences.
There has been much recent progress, especially, on the representation theory of finite groups of Lie type, which ties together the Lie theory and group theory themes of the program. Modular representation theory is also an area of considerable activity.