As the transmission capacity and distance increase, Kerr nonlinearity becomes more and more obvious, which is currently becoming a major factor limiting the fiber transmission performances. The basic principles of some techniques for fiber Kerr nonlinearity compensation in fiber communications are introduced in detail, such as the digital backward propagation algorithm and the perturbation-based nonlinear compensation method. The development status of these technologies is reviewed. The advantages and disadvantages of each algorithm are also presented. Furthermore, the trend and potential applications of these technologies are prospected.