Ramsey's original theorem states that if one finitely colors the $k$ element subsets of ${\Bbb N}$ then there exists an infinite subsequence $M$ of ${\Bbb N}$ all of whose $k$ elements subsets have the same color. This theorem and stronger versions entered into Banach space theory in the 1970's. They were ideal for studying subsequences of a given sequence $(x_i)\subseteq X$ (infinite dimensional separable Banach space). We survey some of these applications and the following problem. $X$ is said to satisfy the ultimate Ramsey theorem if for every finite coloring $(C_i)_{i=1}^n$ of its unit sphere $S_X$ and $\varepsilon>0$ there exists an infinite dimensional subspace $Y$ and $i_0$ so that $S_Y\subseteq (C_{i_0})_\varepsilon =\{x:|x-z|<\varepsilon$ for some $z\in C_{i_0}\}$. What spaces $X$ (if any) have this property? We survey other results including Gowers' block Ramsey theorem for Banach spaces.