Synthetic chemists have developed robust methods to synthesize discrete molecules, linear and branched polymers, and disordered cross-linked networks. However, two-dimensional polymers (2DPs) prepared from designed monomers have been long missing from these capabilities, both as objects of chemical synthesis and in nature. Recently, new polymerization strategies and characterization methods have enabled the unambiguous realization of covalently linked macromolecular sheets. Here we review 2DPs and 2D polymerization methods. Three predominant 2D polymerization strategies have emerged to date, which produce 2DPs either as monolayers or multilayer assemblies. We discuss the fundamental understanding and scope of each of these approaches, including: the bond-forming reactions used, the synthetic diversity of 2DPs prepared, their multilayer stacking behaviors, nanoscale and mesoscale structures, and macroscale morphologies. Additionally, we describe the analytical tools currently available to characterize 2DPs in their various isolated forms. Finally, we review emergent 2DP properties and the potential applications of planar macromolecules. Throughout, we highlight achievements in 2D polymerization and identify opportunities for continued study.