This paper provides a comprehensive survey of the microfoundations, empirical evidence, and estimation issues surrounding the aggregate matching function in labor economics. The matching function is a key tool for modeling frictions in labor markets, such as unemployment and wage inequality, and it captures the relationship between job vacancies and unemployed workers. The paper discusses various microeconomic mechanisms that have been proposed to explain the matching process, including mismatch, coordination failures, worker heterogeneity, and stock-flow matching. It also reviews empirical methods and findings, including the estimation of Beveridge curves, aggregate and sectoral studies, and micro studies. The empirical evidence suggests that the aggregate matching function is well-behaved, with constant returns to scale, and influenced by variables such as unemployment and vacancies. The paper highlights the importance of individual characteristics, such as search intensity and reservation wages, in shaping the matching function. Additionally, it explores the implications of different matching mechanisms, such as ranking and stock-flow matching, and the challenges of aggregating over distinct labor markets. Overall, the paper provides a detailed overview of the current state of research on the matching function, emphasizing its role in understanding labor market dynamics and policy implications.This paper provides a comprehensive survey of the microfoundations, empirical evidence, and estimation issues surrounding the aggregate matching function in labor economics. The matching function is a key tool for modeling frictions in labor markets, such as unemployment and wage inequality, and it captures the relationship between job vacancies and unemployed workers. The paper discusses various microeconomic mechanisms that have been proposed to explain the matching process, including mismatch, coordination failures, worker heterogeneity, and stock-flow matching. It also reviews empirical methods and findings, including the estimation of Beveridge curves, aggregate and sectoral studies, and micro studies. The empirical evidence suggests that the aggregate matching function is well-behaved, with constant returns to scale, and influenced by variables such as unemployment and vacancies. The paper highlights the importance of individual characteristics, such as search intensity and reservation wages, in shaping the matching function. Additionally, it explores the implications of different matching mechanisms, such as ranking and stock-flow matching, and the challenges of aggregating over distinct labor markets. Overall, the paper provides a detailed overview of the current state of research on the matching function, emphasizing its role in understanding labor market dynamics and policy implications.