What makes an algebraic curve special?

A survey of special curves, special subvarieties of ${\mathcal{M}}_g$, and related topics. A large portion of the text discusses various possible interpretation of the word ‘special’ in this context by giving also concrete examples. One highlight is the bi-algebraic viewpoint for atypical intersections appearing in Hodge theory as well as, more recently, in Teichmüller theory.