This book is the first systematic and rigorous account of continuum percolation. The authors treat two models, the Boolean model and the random connection model, in detail, and discuss a number of related continuum models. Where appropriate, they make clear the connections between discrete percolation and continuum percolation. All important techniques and methods are explained and applied to obtain results on the existence of phase transitions, equality of certain densities, continuity of critical densities with respect to distributions, uniqueness of the unbounded component, covered volume fractions, compression, rarefaction and so on. The book is self-contained, assuming only familiarity with measure theory and basic probability theory. The approach makes use of simple ergodic theory, but the underlying geometric ideas are always made clear.