The construction of finitely generated infinite torsion groups (especially of bounded exponent), i.e., the Burnside problem. Schur, building on work of Burnside, proved finitely generated torsion linear groups were finite. Golod constructed the first finitely generated infinite torsion group (a p-group with unbounded exponent) using the Golod-Shafarevich theorem. Adian and Novikov proved that finitely generated free groups of exponent n are infinite for n sufficiently large and odd. Relatively simple to understand examples of finitely generated groups (with unbounded exponent) were constructed by Grigorchuk, Suschanskii, Gupta-Sidki using groups acting on rooted trees (or automata) (I believe Aleshin first suggested using automata for the Burnside problem, but I don't know if he proved any of his constructions worked).