Laser plasma accelerator can accelerate electron beams to GeV level within a few centimeters, which provides a driving source for the development of table-top free electron laser. However, due to the difficulties in laser plasma acceleration and the development of existing technology, the quality of electron beam is not high enough to realize free electron laser, especially in the aspects of stability, divergence and energy spread of electron beam, which present substantial obstacles to the development of compact free electron laser. In this paper, the latest research progress of free electron laser based on laser plasma accelerator is introduced, the main challenges in the experiment for high gain free electron laser are analyzed, then corresponding solutions and experimental progress are summarized, and the future development is prospected. A recent research shows that the high-quality electron beam produced by controlling and optimizing the injection and acceleration of laser plasma accelerator can achieve spontaneous emission amplification and produce high gain radiation in the exponential gain regime, which promotes the research for free electron laser based on laser plasma accelerator to a new stage.