By moving from massive antennas to antenna surfaces for software-defined wireless systems, the reconfigurable intelligent surfaces (RISs) rely on arrays of unit cells to control the scattering and reflection profiles of signals, mitigating the propagation loss and multipath attenuation, and thereby improving the coverage and spectral efficiency. In this paper, covert communication is considered in the presence of the RIS. While there is an ongoing transmission boosted by the RIS, both the intended receiver and an eavesdropper individually try to detect this transmission using their own deep neural network (DNN) classifiers. The RIS interaction vector is designed by balancing two (potentially conflicting) objectives of focusing the transmitted signal to the receiver and keeping the transmitted signal away from the eavesdropper. To boost covert communications, adversarial perturbations are added to signals at the transmitter to fool the eavesdropper's classifier while keeping the effect on the receiver low. Results from different network topologies show that adversarial perturbation and RIS interaction vector can be jointly designed to effectively increase the signal detection accuracy at the receiver while reducing the detection accuracy at the eavesdropper to enable covert communications.