Let $K$ be a field with a valuation $v$. Given a projective variety $X$ over $K$, we can associate an analytification $X^{an}$ with respect to $v$ called the Berkovich space. These spaces appear in various contexts, such as tropical geometry and number theory. More recently there have appeared surprising connections between Berkovich geometry and birational geometry. I will give a brief overview of Berkovich spaces with examples, and describe how the birational geometry of $X$ is reflected in the geometry of the associated Berkovich space.