This talk was aimed at an audience dominated by algebraic geometers. Because my funding for 2005-2006 came from the algebraic geometry Research Training Grant, I was asked to give an overview of what I had been working on during the past year.

I begin by describing why algebraic geometers care about cohomology and p-adic methods, and then give an overview of three different p-adic cohomology theories: crystalline, Monsky-Washnitzer, and rigid. I discuss how these theories are related to eachother and what motivated their development. I describe rather explicitly the construction of MW cohomology.

I discuss certain compatibility issues that frequently arise when studying p-adic cohomology, in particular Frobenius compatibility. I then pose a specific compatibility question that is a generalization of one I had to solve in the course of my thesis work.