The aggregate matching (hiring) function relates gross hires to labor market tightness. Decompositions of aggregate hires show how the hiring process differs across different groups of workers and of firms. Decompositions include employment status in the previous month, age, gender and education. Another separates hiring between part-time and full-time jobs, which show different patterns in the current recovery. Shift-share analyses are done based on industry, firm size and occupation to show what part of the residual of the aggregate hiring function can be explained by the composition of vacancies. The hiring process appears to shift as a recovery starts, coinciding with shifts in the Beveridge curve. The paper also discusses some issues in the modeling of the labor market.