Phase retrieval is a technique used to recover a signal or image from phaseless measurements. It has important applications in fields such as optics, X-ray crystallography, and astronomy, where phase information is often lost during the measurement process. In this talk, we will introduce several recent phase retrieval algorithms and discuss their underlying theory. We will first focus on algorithms that employ spectral initialization, demonstrating that the randomized Kaczmarz method achieves a linear convergence rate for complex-valued signals, while the Gauss-Newton method attains a quadratic convergence rate without requiring sample splitting. We will then turn to algorithms initialized randomly. Through geometric landscape analysis of nonconvex optimization problems, we show that gradient descent with a random initial guess converges to the global solution with high probability.