  Questions of `how best to acquire data' are essential to modeling and
prediction in the natural and social sciences, engineering applications, and
beyond. Optimal experimental design (OED) formalizes these questions and
creates computational methods to answer them. This article presents a
systematic survey of modern OED, from its foundations in classical design
theory to current research involving OED for complex models. We begin by
reviewing criteria used to formulate an OED problem and thus to encode the goal
of performing an experiment. We emphasize the flexibility of the Bayesian and
decision-theoretic approach, which encompasses information-based criteria that
are well-suited to nonlinear and non-Gaussian statistical models. We then
discuss methods for estimating or bounding the values of these design criteria;
this endeavor can be quite challenging due to strong nonlinearities, high
parameter dimension, large per-sample costs, or settings where the model is
implicit. A complementary set of computational issues involves optimization
methods used to find a design; we discuss such methods in the discrete
(combinatorial) setting of observation selection and in settings where an exact
design can be continuously parameterized. Finally we present emerging methods
for sequential OED that build non-myopic design policies, rather than explicit
designs; these methods naturally adapt to the outcomes of past experiments in
proposing new experiments, while seeking coordination among all experiments to
be performed. Throughout, we highlight important open questions and challenges.
