The proof of the Weil conjectures by Grothendieck and Deligne relies on the powerful machinery of étale cohomology. While étale cohomology provides deep structural insight, it is not well-suited for explicit computations. In response, a variety of p-adic cohomology theories have been developed that are better for calculations. In this talk, I will introduce Monsky–Washnitzer cohomology, establish some trace formulas, and illustrate how p-adic methods can be used for effective computation of zeta functions of hyperelliptic curves and K3 surfaces.