Tate's work on Rigid Analytic Spaces can be used to obtain the
$p$-adic uniformization of a curve. In this talk, I will describe a
criterion determining which hyperelliptic curves admit this type of
uniformization. Then, we will discuss Mumford curves, which are the
uniformizing spaces, and explain how to approximate the $p$-adic
uniformization of a given totally split hyperelliptic curve.