We identify knowledge spillovers from foreign investment to domestic firms using novel measures of ``closeness'' of foreign-owned and domestic and domestic firms in product space and in technology space. We rely on a new data set that spans six advanced countries connecting firms internationally in order to, a) separate competition effects on domestic firms from knowledge spillovers when domestic and foreign-owned firms are close in product space, and b) identify spillovers from foreign-owned firms that are close to domestic firms in technology space. We find strong negative competition effects on domestic firms that produce in the same {four-digit} sector as the foreign firms and positive knowledge spillovers to domestic firms operating in the same {two-digit} sector (but in different four-digit sectors). Using a measure of ``technological closeness, '' building on the work of Bloom, Schankerman, and van Reenen (2013), we find significant knowledge spillovers to firms which are close in technology space. On average, knowledge spillovers explain 60 percent of the total factor productivity improvement for domestic firms that are technologically close to foreign firms.