This book is the English translation of a French version published previously by HERMANN (Éditeurs des Sciences et des Arts, Paris). The classical problem of moduli spaces of curves is generalized here to the problem of a description of the moduli space of curves of the same equisingularity class. The first four chapters are devoted to equisingularity invariants, parametrization of the moduli space and its detailed description (in specific cases). After a review of specific examples, the author studies the generic component of moduli space with a particular characteristic. The last third of the book contains an appendix written by B. Tessier. The reader will find a version of the ‘product decomposition’ theorem and a natural compactification of the moduli space of a plane branch with a given characteristic.