Recently, lots of attention and researches have been paid to the viscoelastic instabilities which are generated by the interaction between the elastic stress and inertial stress and even can emerge at extremely low Reynolds number (Re) different from Newtonian instabilities. When the elasticity number (defined as the ratio between Deborah number and Reynolds number) is significant, the purely elastic instabilities would happen at Re << ReC, where ReC is the critical Re of inertial instabilities i.e. the inertial stress is negligible. Generally, the visoelastic fluids are characterized by a complicated rheological properties such as memory properties, shear-thinning and non-zero normal stress difference and so on. Nonetheless, based on various experiments, theoretical analysis and numerical simulation, general understanding of the phenomenon has been achieved. In the paper we make a review on the development of the research on purely elastic instabilities mainly from the experimental and theoretical viewpoint. It is organized as the following categories：the introduction of the viscoelastic fluids, the development and results obtained from experiments (such as Taylor-Couette flows, cone-and-plate flows and plate-and-plate flows), and the development and results from theoretical analysis (such as linear and nonlinear stability analysis, direct numerical simulation, gradient energy theory and so on). Then summaries of the purely elastic instability mechanism, evolvement and the influencing factors are achieved based on the results from experiments and theoretical analysis. In the end we proposed the existing problems and suggestions for future work.