In this talk, I will outline the proof that $r(4,t) = t^{3 - o(1)}$, which solves a long-standing open problem of Erdos. The proof combines a variety of techniques, and lends itself to improving bounds on various other Ramsey numbers $r(F,t)$ where $F$ is a fixed graph., as well as applications to a variety of extremal and Ramsey problems.