  For every positive integer h, the representation function of order h
associated to a subset A of the integers or, more generally, of any group or
semigroup X, counts the number of ways an element of X can be written as the
sum (or product, if X is nonabelian) of h not necessarily distinct elements of
X. The direct problem for representation functions in additive number theory
begins with a subset A of X and seeks to understand its representation
functions. The inverse problem for representation functions starts with a
function f:X ->N_0 U {\infty} and asks if there is a set A whose representation
function is f, and, if the answer is yes, to classify all such sets. This paper
is a survey of recent progress on the inverse representation problem.
