We quantify the role of gender-specific firm wage premiums in explaining the private-sector gender gap in hourly wages using a harmonized research design across 11 matched employer-employee datasets --- ten European countries and Washington State, USA. These premiums explain the gender gap when women are less likely to work at high-paying firms (sorting) or receive lower premiums than men within the same firm (pay-setting). We find that firm wage premiums account for a substantial share of variation in gender wage gaps, ranging from 15 to 32 percent. While both mechanisms matter, sorting is the predominant driver of the firm contribution to the gender wage gap in most countries. Three patterns are broadly consistent: (1) women sorting into lower-paying firms becomes increasingly pronounced with age; (2) women are more concentrated in low-paying firms with a high share of part-time workers; and (3) pay-setting gaps are largest in high-wage firms, where women receive about 90 percent of the rents men receive from firm surplus gains.