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A subset of omega is a base for G if its pointwise stabiliser in G is trivial. In this paper we introduce and study an associated graph sigma(G), which we call the Saxl graph of G. The vertices of sigma(G) are the points of omega, and two vertices are adjacent if they form a base for G. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of sigma(G) for a finite transitive group G, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if G is a primitive group with a base of size 2, then the diameter of sigma(G) is at most 2. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when G = S-n or A(n) (with n > 12) and the point stabiliser of G is a primitive subgroup. 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