Integration of individual two-dimensional graphene sheets into macroscopic structures is essential for the application of graphene. A series of graphene-based composites and macroscopic structures have been recently fabricated using chemically derived graphene sheets. However, these composites and structures suffer from poor electrical conductivity because of the low quality and/or high inter-sheet junction contact resistance of the chemically derived graphene sheets. Here we report the direct synthesis of three-dimensional foam-like graphene macrostructures, which we call graphene foams (GFs), by template-directed chemical vapour deposition. A GF consists of an interconnected flexible network of graphene as the fast transport channel of charge carriers for high electrical conductivity. Even with a GF loading as low as ∼ 0.5 wt%, GF/poly(dimethyl siloxane) composites show a very high electrical conductivity of ∼10 S cm^-1, which is ∼6 orders of magnitude higher than chemically derived graphene-based composites. Using this unique network structure and the outstanding electrical and mechanical properties of GFs, as an example, we demonstrate the great potential of GF/poly(dimethyl siloxane) composites for flexible, foldable and stretchable conductors.