the book collects results about realization spaces of polytopes. it gives a presentation of the authors universality theorem for 4-polytopes. it is a comprehensive survey of the important results that have been obtained in that direction. the approaches chosen are direct and very geometric in nature. the book is addressed to researchers and to graduate students. the former will find a comprehensive source for the above mentioned results. the latter will find a readable introduction to the field. the reader is assumed to be familiar with basic concepts of linear algebra.