This article presents a literature review of published methods for damage identification and prediction in mechanical structures. It discusses ways which can identify and predict structural damage from dynamic response parameters such as natural frequencies, mode shapes, and vibration amplitudes. There are many structural applications in which dynamic loads are coupled with thermal loads. Hence, a review on those methods, which have discussed structural damage under coupled loads, is also presented. Structural health monitoring with other techniques such as elastic wave propagation, wavelet transform, modal parameter, and artificial intelligence are also discussed. The published research is critically analyzed and the role of dynamic response parameters in structural health monitoring is discussed. The conclusion highlights the research gaps and future research direction.