Low probability of detection (LPD) is an extremely important characteristic of an underwater acoustic communication (UWAC) system when used for military-related applications, since the detection of a communication signal in the channel may reveal the presence of the transmitter or receiver. Furthermore, the recent advances in the understanding of the environmental effects of sound transmission in the ocean have led to a growing interest in LPD for UWAC also for civilian use. This is because systems that are designed for reliable communication at low signal power have a reduced environmental impact. In this paper, we identify the main challenges for the design of UWAC LPD systems. We describe and classify common approaches for transmission, reception, and interception of LPD signals, and we discuss their advantages and weaknesses. We also present several methods to determine the LPD capability of a system and suggest to adopt the range ratio test as a performance measure that captures the effects of signal propagation through the UWAC channel and the capabilities of the communication receiver and a signal interceptor. In light of the environmental benefits of LPD transmission and ongoing discussions about limiting the power spectral density of UWAC signals through regulations, we believe that LPD transmission is an area of growing importance for UWAC research and development. We hope that this paper serves as a motivation and a starting point for further research in this field.