If $f(x)$ is a polynomial with integer coefficients, a {\it periodic point} of period $k$ over $\mathbb{C}$
is a root of the polynomial $f^k(x)-x$. In this talk, we present results on the existence of periodic points
for polynomials modulo primes; in particular, on average modulo how many primes should a given polynomial
have a periodic point of exact period $k$? We show that, in fact, under rather weak hypotheses, the answer
is at most $1/k$. We will then give an infinite family of quadratic polynomials that satisfy these hypotheses
for infinitely many $k$.