Comment: 42 pages
v4: modified exposition in the main text + minor changes and additionsWe analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that almost every game that is 'generic' (without indifferences) and has a pure Nash equilibrium and a 'large' number of players is connected, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in almost every large generic game that has one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in every generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.