The resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a ( unit) resistor. Let G + H, G x H, G[H], G circle times H, G boxed times H, G(degrees)H and G{H} be the join, product, composition, direct product, strong product, corona and rooted product of two graphs G and H, respectively. In this paper, formulae for resistance distances of these composite graphs are given in terms of parameters of the parent graphs, and some properties are established. Explicit formulae are obtained for resistance distances of some classes of graphs, including rook graphs, diagonal mesh graphs and generalized double graphs.