Non-destructive testing using acoustic waves has emerged as a potential high-impact technology for improving the safety of elastic structures and reducing life-cycle costs. Although this technology has seen a number of significant developments, normal operational conditions, such as applied stresses, may hamper its reliability in terms of defects detection. In this context, the nonlinear interactions between the mechanical stress in the materials and the acoustic waves have been studied over the years. The influence of applied stress on acoustic wave propagation is commonly termed acoustoelasticity. It is worth noting that the acoustoelastic effect property is deeply related to the acoustic modes propagation, direction of waves propagation and the material nonlinearity. Thus, understanding the acoustoelastic phenomena may be crucial to fully exploit the benefits of the non-destructive technology. For instance, since damage and applied stress cause changes on wave phase velocity, the damage-induced change should be distinguished from the stress-induced change. Meanwhile, well understanding stress-induced changes can yield the opportunities of stress measurement. To the best of the author's knowledge, there is no review on the subject of acoustoelastic effects. This paper reviews most of the corresponding studies published in recent years, with an emphasis on non-destructive testing and structural health monitoring applications. The main objective of this review paper is to collate the research performed in the area of acoustoelastic effects during the last 23 years, thereby giving a broad perspective on the state of the art in this research field. Moreover, some disputed and obscure understandings of the acoustoelastic effects are discussed, and the relationship between the initial stress and the material symmetry is examined.