All wireless devices expose users to some level of electromagnetic radiation during operation. In many countries, exposure levels are strictly regulated to ensure the safety of consumers. Previous research demonstrates that incorporating exposure constraints into transmit signal design leads to substantial capacity gains over traditional power back-off techniques. This is especially vital for millimeter wave systems, which require large array gains to combat high path losses and are more susceptible to a decrease in transmit power. In this work, we present exposure modeling procedures and exposure-aware transmission schemes for millimeter wave systems. We first develop methods to approximate the characteristic matrix of a quadratic model for two exposure measures in the millimeter wave band: incident power density and surface specific absorption rate (SAR). The proposed models can be calculated with a small number of parameters and can be altered to account for changes in the exposure scenario. Software simulations with half-wave dipole antennas corroborate the accuracy of the exposure models in the millimeter wave band. We then exploit the ability of the model to calculate exposure at any point surrounding the device to develop efficient exposure-aware signaling strategies. Finally, we propose a low-complexity perturbation approach to obtain exposure-compliant beamforming vectors. Analytical and numerical results demonstrate that the proposed exposure-aware signaling techniques outperform power reduction approaches.