We study the problem of estimating the states of a linear system based on measured data. We investigate the problem in both deterministic and stochastic settings. In the deterministic case, we develop data-driven conditions under which we can reconstruct state trajectories uniquely. Also, we discuss the case in which we have some missing data in the given input/output measurements. In the stochastic case, we develop a Kalman filter-like algorithm to recursively estimate both states and outputs. Finally, we consider a multi-input multi-output system to elucidate the developed results.