The fault and state estimation problem is addressed for a class of linear discrete time-varying two-dimensional systems subject to state and measurement noises. Two estimators are proposed to compute the estimation of the system state and/or fault recursively, both of which are unbiased with minimum variance. Through formulating the estimation problem as the solvability problem of the corresponding matrix equations of estimator gains and system constraint, the necessary and sufficient condition of the existence and the solution for the proposed estimators are given. An example is used to demonstrate the effectiveness of the proposed estimators.