We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let $$C_1<d< c n/\log ^2 n$$ C 1 < d < c n / log 2 n and let $$\mathcal {M}_{n,d}$$ M n , d be the set of all $$n\times n$$ n × n square matrices with 0 / 1 entries, such that each row and each column of every matrix in $$\mathcal {M}_{n,d}$$ M n , d has exactly d ones. Let M be a random matrix uniformly distributed on $$\mathcal {M}_{n,d}$$ M n , d . Then the smallest singular value $$s_{n} (M)$$ s n ( M ) of M is greater than $$n^{-6}$$ n - 6 with probability at least $$1-C_2\log ^2 d/\sqrt{d}$$ 1 - C 2 log 2 d / d , where c, $$C_1$$ C 1 , and $$C_2$$ C 2 are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form $$M-z\,\mathrm{Id}$$ M - z Id , where $$\mathrm{Id}$$ Id is the identity matrix and z is a fixed complex number.