A really nice example is the (unpublished) work of Larsen and Pink on the "rough" classification of subgroups of $\mbox{GL}_n(k)$. Here's a link: http://www.math.ethz.ch/~pink/ftp/LP5.pdf

In one sentence, the idea is to study these subgroups by looking at their "effective Zariski closures", whereupon techniques of algebraic geometry may be brought to bear on the problem.