The first goal of this series of two talks is to present a very general method for describing a C* algebra as a partial crossed product which often applies to C*-algebras generated by partial isometries. This method is based on a paper I wrote with Marcelo Laca and John Quigg many years ago and it has already been used to describe Cuntz-Krieger algebras, Hecke algebras, algebras associated to integral domains, algebras associated to separated graphs and many others. My second goal is to enlarge the above list by including the

Carlsen-Matsumoto C*-algebras associated to subshifts. This class of algebras was introduced by Matsumoto in 1997 and has a very long and rich history, having been intensively studied by many authors. Nevertheless the theory of partial actions can provide new results, notably a complete characterization of simplicity applying to all subshifts, even those which are not surjective. This part of the talk will be based on a joint paper with M. Dokuchaev.