This paper explores the impact of trade on growth when firms are heterogeneous. We find that greater openness produces anti- and pro-growth effects. The Hopenhayn-Melitz-model selection effects raises the expected cost of introducing a new variety and this tends to slow the rate of new-variety introduction and hence growth. The pro-growth effect stems from the impact that freer trade has on the marginal cost of innovating. The balance of the two effects is ambiguous with the sign depending upon the exact nature of the innovation technology and its connection to international trade in goods and ideas. We consider five special cases (these include the Grossman-Helpman, the Coe-Helpman, and the Rivera-Batiz-Romer models) two of which suggest that trade harms growth; the others predicting the opposite.