For over a century since the Nobel prize winning work by Einstein (1905), atomic photoionization was thought to be an instantaneous process. Recent experimental progress in ultrashort laser pulse generation has allowed to resolve this process in time. The concept of time delay introduced by Wigner (1955) in particle scattering appears to be central to the time resolution of photoionization. In this review, we examine the fundamental concepts of time-respolved atomic ionizataion processes. We will follow the recent literature and show how the initial disagreements between theory and experiment, which persisted for nearly a decade, were finally reconciled. We will also outline the exciting prospects of this field driven by modern experimental and computational technologies.