Abstract: SPH (smoothed particle hydrodynamics) is a gridlessLagrangian numerical method based on kernel approximation. In this method,the continuum equations of fluid dynamics are replaced by particleequations. SPH can deal with large deformation and extensive tangling, whichare difficult to be handled in FEM, as well as free surface and materialinterface, which are tricky problems in FDM. SPH is robust in the simulation ofimpact, explosion and crack. However, the tensile instability is its biggestdrawback in the applications to solid dynamics. Von Neumannstability analysis shows that the criterion for stability or instabilitycan be expressed in terms of the stress state and the second derivative of the kernel function.At present, SPH tensile instability is alleviated in the stress point method,artificial stress method, CSPM (corrected smoothed particle hydrodynamicsmethod), conservative smoothing method and other methods, but they can not completely overcome SPH tensile instability. In thispaper, the theory of SPH and the idea of Von Neumann stability analysis areintroduced, and the research results and recent advances in SPH tensileinstability are analyzed. The existing problems and future trends inthese fields are discussed.