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ABSTRACT: Entanglement of quantum states is one of the most fundamental concepts in quantum information theory, due to its usefulness in various quantum protocols and computations. In this talk, we introduce a mathematical approach to quantum entanglement using the probabilistic method, mainly on random matrix theory, combined with high-dimensional convex analysis. In particular, we will discuss the threshold phenomena in quantum entanglement: the generic entanglement behavior of random quantum states "flips" to its opposite when an intermediate parameter crosses a critical threshold value. Finally, we will present our recent work with Ion Nechita, which describes threshold phenomena in random quantum states with group symmetries.