A translation surface is the datum of an abelian differential on a Riemann surface. Every such pair determines a representation, called the absolute period representation or period character. In the first part of this seminar, we discuss the realization problem for a given representation as the period character of some translation surface, possibly with prescribed data such as the orders of singularities, spin structure, or hyperelliptic structure. In the second part, we focus on the subtle problem of prescribing the so-called relative periods, thereby answering a question posed by Simion Filip.