What can be computed with bounded time, space or randomness, and the lower bounds that separate classes.
Algorithms for quantum computers, from Shor's factoring and Grover's search to quantum simulation, and the evidence about which speedups are real, provable or only heuristic.
Algorithms that multiply n×n matrices with fewer than n³ arithmetic operations, from Strassen (1969) to the laser method, and their link to tensor rank.