The full configuration interaction quantum monte carlo (FCIQMC) method is a sparse and highly parallelizable CI eigensolver in quantum chemistry. Relying on the sparsity of the CI Hamiltonian and its eigensolutions, FCIQMC is in general not limited by the active space sizes, in contrast to conventional CI methods, which are limited to at most 18 electrons and 18 orbitals. Due to the nearly linear parallelization of the algorithm, FCIQMC takes full advantage of modern computer architectures, and allows a large number of electrons and orbitals to be explicitly correlated; active spaces containing up to 96 electrons in 159 orbitals have been reported to date. The FCIQMC dynamics also allows to stochastically sample reduced density matrices (RDMs) via the replica trick, enabling the calculation of properties and orbital optimization, in the form of the stochastic complete and generalized active space self-consistent field (Stochastic-CASSCF and Stochastic-GASSCF, respectively) methods. In FCIQMC, the stochastically sampled wave function is represented by walkers distributed across the electronic configurations that form the CI vector space. The CI coefficient of a given configuration is proportional to the expectation value of the walker occupation on that configuration averaged over time, c i ∝ N w i τ . In the limit of large walker numbers, the approximated c i values become exact. Electronic configurations corresponding to vanishingly small c i coefficients are never or only rarely occupied by walkers. Since only occupied configurations have an impact on the computational costs of a FCIQMC simulation, the method relies on and takes advantage of sparsity. The initiator approximation (i-FCIQMC) and its further refinement, the adaptive shift method, increases the sparsity of the wave function and greatly improve the convergence with respect to the walker number.