  The development of control methods based on data has seen a surge of interest
in recent years. When applying data-driven controllers in real-world
applications, providing theoretical guarantees for the closed-loop system is of
crucial importance to ensure reliable operation. In this review, we provide an
overview of data-driven model predictive control (MPC) methods for controlling
unknown systems with guarantees on systems-theoretic properties such as
stability, robustness, and constraint satisfaction. The considered approaches
rely on the Fundamental Lemma from behavioral theory in order to predict
input-output trajectories directly from data. We cover various setups, ranging
from linear systems and noise-free data to more realistic formulations with
noise and nonlinearities, and we provide an overview of different techniques to
ensure guarantees for the closed-loop system. Moreover, we discuss avenues for
future research that may further improve the theoretical understanding and
practical applicability of data-driven MPC.
