Water hammer or flow transients occur due to a sudden variation (over time) in flow properties that can lead to pipe system failure or damage. In addition, research on wave propagation and signal processing theories has developed different ways to detect anomalies in pipe systems. The two developments concerning pipe system safety and damage localization are of essential need in viscoelastic (VE) pipes, as their application in various industries is growing, given their favorable mechanical properties. With no literature review focusing on the topic, this paper aims to fill the current literary gap on transient waves in VE pipes. It highlights developments in the research field and elaborates on relevant water hammer concepts in VE pipes, including mathematical modeling, experimental setups, numerical solutions, parameter calibration, defect detection, and surge control. The comprehensive review concludes that a reliable transient wave model in viscoelastic pipes is yet to be fully confirmed despite the significant progress in the recent two decades.