The phase retrieval problem consists of recovering a vector from the pointwise absolute value of its image when acted upon by a linear operator. This problem arises frequently in many physics and engineering applications via the context of Fourier transform operators, and has also been recently studied in the context of random Gaussian matrices and other classes of linear operators. This talk shall present the mathematical foundations of the the phase retrieval problem, as well as some recently developed algorithms which apply techniques from convex and nonconvex optimization and semidefinite programming.