Police agencies devote vast resources to minimising the time that it takes them to attend the scene of a crime. Despite this, the long-standing consensus is that police response time has no meaningful effect on the likelihood of catching offenders. We revisit this question using a uniquely rich dataset from the Greater Manchester Police. To identify causal effects, we use a novel strategy that exploits discontinuities in distance to the response station across locations next to each other, but on different sides of division boundaries. Contrary to previous evidence, we find large and strongly significant effects: in our preferred estimate, a 10% increase in response time leads to a 4.7 percentage points decrease in the likelihood of clearing the crime. We find stronger effects for thefts than for violent offenses, although the effects are large for every type of crime. We find suggestive evidence in support of two mechanisms: the likelihood of an immediate arrest and the likelihood that a suspect will be named by a victim or witness both increase as response time becomes faster. We argue that, under conservative assumptions, hiring an additional response officer would generate a benefit, in terms of future crime prevented, equivalent to 170% of her payroll cost.