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In 1999 Chas and Sullivan showed that the homology of the free loop space ofan oriented manifold admits the structure of a Batalin-Vilkovisky algebra. Inthis paper we give a complete description of this Batalin-Vilkovisky algebrafor complex projective spaces. This builds on a description of the ringstructure that is due to Cohen, Jones and Yan. In the course of the proof weestablish several new general results. These include a description of howsymmetries of a manifold can be used to understand its string topology, and arelationship between characteristic classes and circle actions on spherebundles.

String topology for complex projective spaces