Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. When used as an emulator, it belongs to a general scheme called reduced basis methods. But it can also be used to extrapolate solutions to the quantum many-body problem for parameter values where solutions are otherwise not computable.