This thesis examines algorithmic markets - market mechanisms with algorithms as a core component in their functioning -- and markets with algorithms (that is, canonical markets for algorithmic or data-based goods and services). Our primary focus is on the analysis of these mechanisms and markets in terms of societal concerns such as fairness, privacy, and efficiency (including welfare and revenue). For algorithmic markets, we consider automated ad auctions and call auctions; for markets with algorithms, we examine both an abstract, general data-driven market from a theoretical perspective, and a specific, important data-driven market: the U.S. mortgage market. We apply a variety of theoretical tools from various subfields of Computer Science and Economics, including worst-case asymptotic runtime and sample complexity theory, the Probably Approximately Correct (PAC) Learning framework, no-regret learning algorithms, equilibrium analysis, smoothness, differential privacy, and quantitative fairness. In addition to theoretical analysis, we implement various algorithms and mechanisms and perform empirical analysis on real data in relation to the mortgage market. Our results include: new worst-case welfare guarantees, novel equilibrium characterizations, and experimental evaluation of various auction formats in the Ad Types setting; construction and analysis of a differentially private call auction mechanism with good performance and incentive properties; a theoretical analysis elucidating the economic forces that encourage error inequality in data-driven markets; and practical application of quantitative fairness measures to a uniquely rich and exhaustive dataset covering the mortgage market in the United States.