Using the behavioral approach to system theory, we derive a nonparametric representation of linear time-invariant systems based on Hankel matrices constructed from data. The data-driven representation leads to new system identification, signal processing, and control methods. In this talk, we show how the data-driven representation can be used for solving missing data estimation problems. The theory leads to algorithms that are general---can deal simultaneously with missing, exact, and noisy data of multivariable systems---and simple---require basic linear algebra operations only. The results open a practical computational way of doing system theory and signal processing directly from data without identification of a transfer function or a state space representation and doing model-based design. This is joint work with Florian Dörfler (ETH-Zürich).