Consider a rational map F = (F_1,...,F_N) consisting of an N-tuple of rational functions in N variables. The iterates F, F^2, F^3, ... of F determine a dynamical system whose complexity may be measured by the growth of the degree of F^n. A fundamental, and still quite mysterious, invariant is the dynamical degree of F, which is defined to be the limiting value of deg(F^n)^{1/n} as n goes to infinity. Recently people have considered an arithmetic analogue of the dynamical degree in which one looks at the orbit of a point P having rational coordinates and replaces deg(F^n) by the arithmetic size of the coordinates of F^n(P). In this talk I will discuss dynamical degrees, arithmetic degrees, and various results and open problems that relate them. No background in dynamics, algebraic geometry, or number theory will be required for most of the talk.