We consider d-regular directed graphs on n vertices. Every vertex of such graphs has exactly d in-neighbors and d out-neighbors. We show that under some minor restrictions on d, the probability that an adjacency matrix of a random d-regular digraph is singular tends to zero with d growing to infinity. To this end, we establish a few expansion properties of d-regular digraphs, in particular, a Littlewood--Offord type anti-concentration property. 

This is a joint work with A. Litvak, K. Tikhomirov, N. Tomczak-Jaegermann, and P. Youssef