Abstract: A quantitative approach is proposed for stochastic dynamic stability analysis of structures. The classical concept of stochastic dynamic stability is firstly revisited. It is pointed that the dynamic stability of structures not only depends on structural parameters, but also relates to the applied external excitations. A new criterion for identifying dynamic stability of structures is introduced and the definition of stochastic dynamic stability of structures is therefore formulated based on the criterion. According to the principle of preservation of probability, the generalized density evolution equation for probability-preserved system is introduced firstly and then the equation for probability-dissipated system is derived. On the basis, the probability of stability/instability can be obtained via solving the equation for probability-dissipated system by introducing the physical mechanism of dynamic instability of structures as the triggering force of probability dissipation. Numerical algorithms for solving the generalized density evolution equation for probability-dissipated system are provided. According to the obtained probability, it is readily applicable to quantitatively evaluate stochastic dynamic stability of structures in the sense of stability in probability 1 or a given probability. Stochastic dynamic stability analyses of typical structural dynamic systems are carried out by the proposed approach, where the results by Monte Carlo simulations are employed for comparisons. The numerical results verify the e ectiveness of the proposed approach.