We consider covert communication, i.e., hiding the presence of communication from an adversary for multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channels. We characterize the maximum covert coding rate under a variety of settings, including different regimes where either the number of transmit antennas or the blocklength is scaled up. We show that a non-zero covert capacity can be achieved in the massive MIMO regime in which the number of transmit antennas scales up but under specific conditions. Under such conditions, we show that the covert capacity of MIMO AWGN channels converges the capacity of MIMO AWGN channels. Furthermore, we derive the order-optimal scaling of the number of covert bits in the regime where the covert capacity is zero. We provide an insightful comparative analysis of different cases in which secrecy and energy-undetectability constraints are imposed separately or jointly.