Quantum computing has been rapidly evolving as a field, with innovations driven by industry, academia, and government institutions. The technology has the potential to accelerate computation for solving complex problems across multiple industrial sectors. Finance and economics, with many problems exhibiting computationally heavy requirements, comprise a high-profile sector where quantum computing could have a significant impact. Therefore, it is important to identify and understand to what extent the technology could find utility in the sector. This technical review is written for quantum applications researchers, quantitative analysts in finance and economics, and researchers in related mathematical sciences. It is divided into two parts: (i) a survey of quantum algorithms pertinent to problems in finance and economics, and (ii) mapping of several use cases in the sector to the potential quantum algorithms presented in part (i). We discuss some challenges on the pathway to achieving quantum advantage. Ultimately, this review aims to be a catalyst for interdisciplinary research that will accelerate the advent of the practical advantages of quantum technologies to solve complex problems in this sector.