Synthetic two-dimensional polymers (2DPs) are an emerging subclass of 2D materials that are accessible by mild and rational synthesis procedures. Because of their structure, which is reminiscent of a molecular fisherman's net, 2DPs strongly differ from conventional linear polymers. They are expected to show applications in fields such as gas separation, nonlinear optics, and miniaturization of optical devices. The article sketches the historic development that led to the first fully proven representatives, describing each case in detail and comparing the current main access routes. Insights into some application aspects are also given. Alternative synthesis routes are also presented, together with the critical analytical issues. Structure analysis is perhaps the greatest challenge in this field, which arises with regard to proving the periodic structure of a 2DP. This article intends to stimulate the interest of an interdisciplinary community to join forces to develop this promising field even more quickly.