The field of ultracold quantum matter has burgeoned over the last few decades, thanks to the growing capabilities for atomic systems to be probed and manipulated with exquisite control. Researchers can now precisely create and study quantum manybody states that are effectively isolated from the external environment. Much of the work in ultracold matter has focused on systems of alkali or alkaline-earth atoms, mainly due to their ease of cooling. Extending this precise control to molecules has seen rapidly increasing interest and activity, as molecules possess additional degrees of freedom that make them useful for tests of fundamental physics, studying ultracold chemistry and collisions, and engineering qualitatively new types of quantum phases and quantum many-body systems. Here, we review one particularly fruitful research direction: the creation and manipulation of ultracold bialkali molecules. The recent success in creating a quantum gas of polar molecules opens many exciting research opportunities. Atomic, Molecular, and Optical (AMO) systems offer experimentalists the ability to precisely control interactions in a quantum many-body system [1,2], and a rich set of experiments has already been performed. Some prominent examples include studies of the BEC-BCS crossover in two-component Fermi gases [3,4,5] and the superfluid-Mott insulator transition in ultracold bosons loaded into a 3D optical lattice [6]. Neutral atomic systems normally have point-like contact interactions that can be parametrized with a single quantity, the scattering length a, which can be tuned via a Feshbach resonance [2]. In recent years, systems with long-range and anisotropic interactions have garnered much attention. One natural example is the dipolar interaction between polar molecules, which is the focus of this Review. Of course, many other experimental platforms are also being pursued that feature long-range interactions, including highly magnetic atoms [7,8], trapped ions The dipolar interaction exhibits several important attributes. First, the energy scales in dipolar systems are usually much larger than those in typical atomic alkali systems, making it easier to study interaction-driven structural properties and non-equilibrium dynamics. Second, and perhaps more importantly, the anisotropic and long-range nature of the interaction allows for the realization of novel quantum phases, especially when the spatial dimensions of the system can be varied with optical lattices. Some of these engineered systems may find relevance to outstanding problems in condensed-matter physics [13,14]; moreover, qualitatively new types of physics can emerge in the presence of long-range interactions [15,16,17,18,19,20,21]. As a specific example of a unique feature of long-range interactions, a spin-1/2 Hamiltonian can be encoded based on a pair of oppositeparity rotational states where dipolar interactions give rise to a direct spin exchange coupling between molecules [22,23]. With this scheme implemented in an optical lattice, many-body spin dynam...