We analyze recent contributions to growth theory based on the model of expanding variety of Romer [Romer, P. (1990). "Endogenous technological change". Journal of Political Economy 98, 71-102]. In the first part, we present different versions of the benchmark linear model with imperfect competition. These include the "lab-equipment" model, "labor-for-intermediates" and "directed technical change". We review applications of the expanding variety framework to the analysis of international technology diffusion, trade, cross-country productivity differences, financial development and fluctuations. In many such applications, a key role is played by complementarities in the process of innovation.