  In this note we give an overview of various quantities that are used to
measure the complexity of an algebraic dynamical system f:X-->X, including the
dynamical degree d(f), which gives a coarse measure of the geometric complexity
of the iterates of f, the arithmetic degree a(f,P), which gives a coarse
measure of the arithmetic complexity of the orbit of a an algebraic point P in
X, and various versions of the canonical height h_f(P) that provide more
refined measures of arithmetic complexity. Emphasis is placed on open problems
and directions for further exploration.
