  It is a remarkable fact that for many statistics on finite sets of
combinatorial objects, the roots of the corresponding generating function are
each either a complex root of unity or zero. We call such polynomials
\textbf{cyclotomic generating functions} (CGF's). Previous work studied the
support and asymptotic distribution of the coefficients of several families of
CGF's arising from tableau and forest combinatorics. In this paper, we continue
these explorations by studying general CGF's from algebraic, analytic, and
asymptotic perspectives. We review some of the many known examples of CGF's;
describe their coefficients, moments, cumulants, and characteristic functions;
and give a variety of necessary and sufficient conditions for their existence
arising from probability, commutative algebra, and invariant theory. We further
show that CGF's are ``generically'' asymptotically normal, generalizing a
result of Diaconis. We include several open problems concerning CGF's.
