The authors study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinite-dimensional analogues of Axiom A systems. Their main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. They develop a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. They treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems.