The classification of generalized pseudo-Anosov homeomorphisms up to topological conjugacy is a central problem in the dynamics and topology of surfaces, with a long history and a variety of approaches developed by A. Zhirov, L. Mosher, M. Bestvina, M. Handel, J. Los, among others. In this talk, we will present a brief historical overview of the problem and describe an alternative approach, initiated by C. Bonatti, R. Langevin, and E. Jeandenans, based on the use of geometric Markov partitions and their geometric types.

In this context, we will introduce the notion of classification and the different stages involved in this process. We will conclude by showing that the geometric type is a complete invariant for topological conjugacy of generalized pseudo-Anosov homeomorphisms, which allows for their complete classification.