The singularity problem of random matrices asks the probability that a given discrete random matrix is singular. The first such result was obtained by Komlós in 1967. He showed a Bernoulli random matrix is singular with probability o(1). This question can be reformulated for the adjacency matrices of random graphs, either directed or undirected. The most challenging case is when the random graph is sparse. In this talk, I will prove that for random directed and undirected d-regular graphs, their adjacency matrices are invertible with high probability for all d>=3. The idea is to study the adjacency matrices over a finite field, and the proof combines a local central limit theorem and a large deviation estimate.