The basic reproduction number R0 is one of the most important quantities in epidemiology. However, for epidemic models with explicit social structure involving small mixing units such as households, its definition is not straightforward and a wealth of other threshold parameters has appeared in the literature. In this talk I use branching processes to define R0, apply this definition to models with households or other more complex social structures, provide a method for calculating R0 and show inequalities comparing R0 with previous threshold parameters. The comparisons imply that, if R0 > 1, vaccinating a fraction 1 - 1/R0 of the population, chosen uniformly at random, with a perfect vaccine is insufficient to be sure of preventing a large outbreak, and they lead to sharper, easily-computed bounds for the critical vaccination coverage than were previously available. Based on work done jointly with Lorenzo Pellis (Imperial College London) and Pieter Trapman (Stockholm University).