Two-dimensional (2D) materials have been extensively studied in recent years due to their unique properties and great potential for applications. Different types of structural defects could present in 2D materials and have strong influence on their properties. Optical spectroscopic techniques, e.g. Raman and photoluminescence (PL) spectroscopy, have been widely used for defect characterization in 2D materials. In this review, we briefly introduce different types of defects and discuss their effects on the mechanical, electrical, optical, thermal, and magnetic properties of 2D materials. Then, we review the recent progress on Raman and PL spectroscopic investigation of defects in 2D materials, i.e. identifying of the nature of defects and also quantifying the numbers of defects. Finally, we highlight perspectives on defect characterization and engineering in 2D materials.