Numerical methods for solving initial value ODEs, especially large problems arising from semi-discretization of PDEs. Review of Runge-Kutta and multistep methods: consistency, stability, convergence, accuracy. Error estimation and step size control. Stiffness, order reduction, stage order and stiff accuracy. Logarithmic norms and one-sided Lipschitz constants. Symplectic and energy-conserving methods. Monotonicity-, contractility- and positivity-preserving methods. Methods for Multiphysics problems: stabilized methods, exponential, and additive methods. Parallel time integration methods: Parallel, deferred corrections and PFASST, extrapolation.