The minimum wage is often considered a social policy instrument that can help reduce both poverty and welfare receipt. The introduction of the statutory minimum wage in Germany in 2015 provides an interesting case study to analyse not only the potential but also the limitations of minimum wages as an instrument to achieve socially desirable goals such as reduced welfare receipt or poverty. Based on the results of simulation models, descriptive analyses and causal effects studies of the short-term effects, we argue that minimum wages are a rather badly targeted measure when attempting to reduce poverty and welfare receipt.