As a sound transmitting device based on the nonlinear acoustic theory, parametric acoustic array (PAA) is able to generate high directivity and low frequency broadband signals with a small aperture transducer. Due to its predominant technical advantages, PAA has been widely used in a variety of application scenarios of underwater acoustic engineering, such as sub-bottom profile measurement, underwater acoustic communication, and detection of buried targets. In this review paper, we examine some of the important advances in the PAA since it was first proposed by Westervelt in 1963. These advances include theoretical modelling for the PAA, signal processing methods, design considerations and implementation issues, and applications of the PAA in underwater acoustic engineering. Moreover, we highlight some technical challenges which impede further development of the PAA, and correspondingly give a glimpse on its possible extension in the future. This article provides a comprehensive overview of some important works of the PAA and serves as a quick tutorial reference to readers who are interested to further explore and extend this technology, and bring this technology to other application areas.