When I read about mathematical history I hear of breakthroughs. For example, Cartesian geometry, Newton/Leibniz Calculus, and so on. My question is this: What are some recent epoch-making breakthroughs in maths? I'm more interested in theories/new branches of maths such as cartesian geometry and calculus, rather than theorems such as Fermat's Last Theorem and the Riemann Hypothesis.
EDIT: By recent breakthrough's I mean anything after the complete solution to "the riddle of the quintics" by Galois.
EDIT: Examples: Use of computers in mathematics (thanks to Kieren MacMillan), combinatorics, graph theory, etc. I would also appreciate it if you could use your intuition to guess at which new branches of mathematics might appear. For example, imagine you were placed in the mathematical world pre-Newton and knew about everything that was going on at the time, I'm sure it would be possible to anticipate Calculus. Is something like that possible today?