ABSTRACT Young massive clusters (YMCs) are the most intense regions of star formation in galaxies. Formulating a model for YMC formation while at the same time meeting the constraints from observations is, however, highly challenging. We show that forming YMCs requires clouds with densities ≳ 100 cm−3 to collide with high velocities (≳ 20 km s−1). We present the first simulations which, starting from moderate cloud densities of ∼100 cm−3, are able to convert a large amount of mass into stars over a time period of around 1 Myr, to produce dense massive clusters similar to those observed. Such conditions are commonplace in more extreme environments, where YMCs are common, but atypical for our Galaxy, where YMCs are rare.