I will discuss the following ``Algebraicity Conjecture'': A simple group of ``finite Morley rank'' is algebraic. In the first place, I will need to explain what this means, and also (at length) what sort of problems in model theory would lead one to consider a conjecture of this type in the first place. Recent work on this problem involves methods inspired by the techniques used to classify the finite simple groups, along lines suggested by A. Borovik, although the groups under consideration are infinite. Website reference for the ``Borovik program'': http://www.rci.rutgers.edu/~cherlin/FMR