My answer, I fear, will be narrow and obscure, and probably bewildering to most people who are not professional mathematicians. It will also reflect the interests of a particular mathematical specialty, the topology of manifolds.
That said, I am obliged to nominate as the greatest innovation in my field, at least within the past few years, the new methods for analysing topological questions that arise from the study of Ricci flows on Riemannian structures, utilising deep ideas from differential geometry and partial differential equations, that were initially developed by Hamilton, extended by Perelman to produce a proof of the classical Poincare conjecture, extended by Yau and his students, and clarified by Morgan and Tien (to name the most prominent figures).
I apologise if my citation of these arcane matters is opaque to most of your readers and nowhere near as exciting to them as recent work in genetics, neurocognitive science, or even cosmology. It is, however, quite exciting to mathematicians.