An implicit representation of an n-vertex graph G in a family F of graphs assigns to each vertex of G a binary code so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family F but not on the individual graph G. In this talk we will discuss recent developments in the study of implicit representation of graphs, including recent joint work with Hamed Hatami, which refutes the Implicit Graph Conjecture. An implicit representation is called efficient if each vertex is given a code of length O(log n). Every family of graphs admitting such an efficient representation contains at most exp(O(n log(n))) graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph Conjecture (IGC) states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation. In this talk, we show the existence of strong counterexamples to the Implicit Graph Conjecture. In particular, we establish the existence of hereditary graph families with at most factorial speed of growth that require codes of length polynomially large as a function of n.