Complex hyperbolic geometry is a particularly rich field, drawing on Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis. The boundary in complex hyperbolic spaces, known as spherical CR or Heisenberg geometry, reflects this richness. However, while there are a number of books on analysis in such spaces, this book is the first to focus on the geometry, both for complex hyperbolic space and its boundary. Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.