Thin-film organic, colloidal quantum dot, and metal halide perovskite semiconductors are all being pursued in the quest for a wavelength-tunable diode laser technology that does not require epitaxial growth on a traditional semiconductor substrate. Despite promising demonstrations of efficient light-emitting diodes and low-threshold optically pumped lasing in each case, there are still fundamental and practical barriers that must be overcome to reliably achieve injection lasing. This review outlines the historical development and recent advances of each material system on the path to a diode laser. Common challenges in resonator design, electrical injection, and heat dissipation are highlighted, as well as the different optical gain physics that make each system unique. The evidence to date suggests that continued progress for organic and colloidal quantum dot laser diodes will likely hinge on the development of new materials or indirect pumping schemes, while improvements in device architecture and film processing are most critical for perovskite lasers. In all cases, systematic progress will require methods that can quantify how close new devices get with respect to their electrical lasing thresholds. We conclude by discussing the current status of nonepitaxial laser diodes in the historical context of their epitaxial counterparts, which suggests that there is reason to be optimistic for the future.