Real-time electronic structure methods provide an unprecedented view of electron dynamics and ultrafast spectroscopy on the atto- and femtosecond time scale with vast potential to yield new insights into the electronic behavior of molecules and materials. In this Review, we discuss the fundamental theory underlying various real-time electronic structure methods as well as advantages and disadvantages of each. We give an overview of the numerical techniques that are widely used for real-time propagation of the quantum electron dynamics with an emphasis on Gaussian basis set methods. We also showcase many of the chemical applications and scientific advances made by using real-time electronic structure calculations and provide an outlook of possible new directions.