  Recent advances in laser technology enable to follow electronic motion at its
natural time-scale with ultrafast pulses, leading the way towards atto- and
femtosecond spectroscopic experiments of unprecedented resolution.
Understanding of these laser-driven processes, which almost inevitably involve
non-linear light-matter interactions and non-equilibrium electron dynamics, is
challenging and requires a common effort of theory and experiment. Real-time
electronic structure methods provide the most straightforward way to simulate
experiments and to gain insights into non-equilibrium electronic processes. In
this Chapter, we summarize the fundamental theory underlying the relativistic
particle-field interaction Hamiltonian as well as equation-of-motion for
exact-state wave function in terms of the one- and two-electron reduced density
matrix. Further, we discuss the relativistic real-time electron dynamics
mean-field methods with an emphasis on Density-Functional Theory and Gaussian
basis, starting from the four-component (Dirac) picture and continue to the
two-component (Pauli) picture, where we introduce various flavours of modern
exact two-component (X2C) Hamiltonians for real-time electron dynamics. We also
overview several numerical techniques for real-time propagation and signal
processing in quantum electron dynamics. We close this Chapter by listing
selected applications of real-time electron dynamics to frequency-resolved and
time-resolved spectroscopies.
