Quantum manipulation of macroscopic mechanical systems is of great interest in both fundamental physics and applications ranging from high-precision metrology to quantum information processing. A crucial goal is to cool the mechanical system to its quantum ground state. In this review, we focus on the cavity optomechanical cooling, which exploits the cavity enhanced interaction between optical field and mechanical motion to reduce the thermal noise. Recent remarkable theoretical and experimental efforts in this field have taken a major step forward in preparing the motional quantum ground state of mesoscopic mechanical systems. This review first describes the quantum theory of cavity optomechanical cooling, including quantum noise approach and covariance approach; then the up-to-date experimental progresses are introduced. Finally, new cooling approaches are discussed along the directions of cooling in the strong coupling regime and cooling beyond the resolved sideband limit. ; Comment: Invited review paper, 32 pages, 10 figures