Due to the broadcast nature of millimeter wave (mmWave) communications, physical layer security has always been a fundamental but challenging concern. Fortunately, the recent advance of intelligent reflecting surface (IRS) introduces another dimension for mmWave secure communications by reconfiguring the transmission environments. In this article, we investigate the physical layer security issue for IRS-aided mmWave communications. Specifically, two scenarios are considered, i.e., with and without the knowledge of the eavesdropper's channel. When the eavesdropper's channel is known, under the unit modulus constraints at the IRS and the transmit power constraint at the access point (AP), the secrecy rate is maximized by jointly designing the beamforming vector of the AP and the phase shifts of the IRS. We propose an alternating optimization algorithm for improving the secrecy rate to solve the non-convex optimization problem. In particular, the phase shift matrix of the IRS is firstly optimized based on the manifold optimization algorithm. Under the given phase shift matrix of the IRS, the original problem is transformed into a difference of convex (DC) programming problem. To solve this problem, we use the successive convex approximation (SCA)-based method to transform the original problem into a convex approximation problem, and the transmit beamforming vector is obtained by solving the convex approximation problem. In addition, when the eavesdropper's channel is unknown, under the transmit power constraint and the minimum user rate constraint, an artificial-noise (AN)-aided scheme is proposed to jam the eavesdropper by maximizing the AN power. Numerical results evaluate that the performance of our proposed schemes is better than that of the conventional scheme in terms of secrecy rate. Moreover, simulations also demonstrate that the secrecy behaviors of the IRS-aided mmWave communication system are superior to the mmWave communications without IRS.