This paper develops a theory of firm selection and growth and embeds it into an international trade framework of balanced growth. I assume that firm-level growth is the result of idiosyncratic productivity improvements while there is continuous arrival of new potential producers. Firms can also pay an increasing market penetration cost to sell more to a given market. The model is consistent with a set of salient regularities of firm and exporter selection and growth, as well as the observed distribution of sales. I calibrate the model parameters that determine firm dynamics by looking at the exit rates of a cohort at the US census and the elasticity of trade in Eaton and Kortum. This elasticity is regulated in the model by the firm-level growth process. The calibrated model can account for almost all the turnover and growth of US census cohorts over two decades. It can also account for a large part of the turnover and growth of Colombian exporters in individual destinations.