Suppose $T: X \rightarrow X$ is a map of a metric space preserving a probability measure $\mu$ and $\phi: X\rightarrow R$ is a H\"older observation on $X$. We may define $M_n (x)=\max \{ \phi(x), \phi ( T x), \ldots , \phi (T^n x) \}$, the sequence of successive maxima. Extreme value theory is concerned with the existence and type of nondegenerate distributions $G(v)$ obtained by scaling $M_n$ by constants $a_n>0,b_n$ in the sense that $\mu (a_n (M_n -b_n)\le v )\rightarrow G(v)$. For iid random variables there are only three possible limiting distributions, Types I, II and III. This is a similar phenomena to the universality of the central limit theorem. Extreme value theory has implications for hitting and return time statistics by taking $\phi$ to be a function monotonically decreasing as a function of distance from a distinguished point $x_0\in X$, for example $\phi (x)=-\log d(x,x_0)$. We describe recent results on extreme value theory for certain classes of nonuniformly expanding maps, hyperbolic billards, lozi-type maps and suspension flows. Some of this work is joint with Chinmaya Gupta, Mark Holland and Andrew Torok.