Period mappings and differential equations. From $\mathbf{C}$ to ${\mathbf{C}}_{\mathbf{p}}$

This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential equations, group theory and geometry. The text ends with a detailed discussion of $p$-adic triangle groups.