The focus of this talk will be on computing the point counts of algebraic varieties, i.e., number of solutions of a system of polynomial equations over finite fields. The zeta function encodes the point counts over an infinite tower of finite field extensions and enjoys the property of being a rational function over $\mathbb{Q}$. Further, the zeta function can be recovered from certain invariants of the variety in question, using an appropriate cohomology theory. I will review the state of the art on efficient algorithms to compute the zeta function of varieties, including the dimension one case of curves (covering the works of Schoof, and Pila) and report on our generalisations for the first cohomology (joint work with Diptajit Roy and Nitin Saxena) and ongoing work on the second cohomology, which addresses a question of Edixhoven.