For varieties over finite fields, counts of points on the variety are intimately related to cohomology. For a variety defined over a more general base, it is natural to ask what conclusions we can draw about cohomology if the point counts over finite fields (obtained by base changing) are polynomial in the size of the finite field. In this talk, we will explore the rich web of theorems that relate point counting and cohomology. We will also see some examples that illustrate both the power and the subtleties of these results.