Variational data assimilation (VAR) is described in its various forms and their mathematical formulations are explained, including three-dimensional/four-dimensional VAR, first guess at appropriate time (FGAT), Physical-space Statistical Analysis System (PSAS), and incremental approaches. A historical overview of and differences in the calculus of variations and optimal control theory, the root theories on VAR, are also discussed, which are represented by the Euler–Lagrange equations and Pontryagin’s maximum (minimum) principle, respectively. Furthermore, major elements of VAR are reviewed with an emphasis on various formalisms of cost function, including Tikhonov regularization, strong- versus weak-constraint and incremental formulation, and on specification and diagnosis of error covariances, including observation error covariance, background error covariance, and model error covariance. Issues on minimization of the VAR cost function, including gradient, preconditioning, and assimilation period, are also addressed.