Due to a changing climate and increased urbanization, an escalation of urban flooding occurrences and its aftereffects are ever more dire. Notably, the frequency of extreme storms is expected to increase, and as built environments impede the absorption of water, the threat of loss of human life and property damages exceeding billions of dollars are heightened. Hence, agencies and organizations are implementing novel modeling methods to combat the consequences. This review details the concepts, impacts, and causes of urban flooding, along with the associated modeling endeavors. Moreover, this review describes contemporary directions towards urban flood resolutions, including the more recent hydraulic-hydrologic models that use modern computing architecture and the trending applications of artificial intelligence/machine learning techniques and crowdsourced data. Ultimately, a reference of utility is provided, as scientists and engineers are given an outline of the recent advances in urban flooding research.