Using the March Current Population Survey data from 1984 to 2013, I provide a comprehensive evaluation of how minimum wage policies influence the distribution of family incomes. I find robust evidence that higher minimum wages shift down the cumulative distribution of family incomes at the bottom, reducing the share of non-elderly individuals with incomes below 50, 75, 100, and 125 percent of the federal poverty threshold. The long run (3 or more years) minimum wage elasticity of the non-elderly poverty rate with respect to the minimum wage ranges between -0.22 and -0.55 across alternative specifications that subsume most of the approaches used in the literature to construct valid counterfactuals. Inverting the policy's effect on the cumulative distribution, I estimate minimum wage elasticities for unconditional quantiles of family incomes. The long run minimum wage elasticities for the 10th and 15th unconditional quantiles of equivalized family incomes range between 0.15 and 0.49 depending on specification. A reduction in public assistance partly offsets these income gains, which are on average 72% as large when using an expanded income definition including tax credits and non-cash transfers.