We present cosmological hydrodynamical simulations of the formation of dwarf galaxies in a representative sample of haloes extracted from the Millennium-II Simulation. Our six haloes have a z= 0 mass of ∼1010 M⊙ and show different mass assembly histories which are reflected in different star formation histories. We find final stellar masses in the range 5 × 107– 108 M⊙, consistent with other published simulations of galaxy formation in similar mass haloes. Our final objects have structures and stellar populations consistent with observed dwarf galaxies. However, in a Λ cold dark matter universe, 1010 M⊙ haloes must typically contain galaxies with much lower stellar mass than our simulated objects if they are to match observed galaxy abundances. The dwarf galaxies formed in our own and all other current hydrodynamical simulations are more than an order of magnitude more luminous than expected for haloes of this mass. We discuss the significance and possible implications of this result.