We consider the Hausdorff dimension of random covering sets formed by balls with centres chosen independently at random according to an arbitrary Borel probability measure on Rd$\mathbb {R}<^>d$ and radii given by a deterministic sequence tending to zero. We prove, for a certain parameter range, the conjecture by Ekstr & ouml;m and Persson concerning the exact value of the dimension in the special case of radii (n-alpha)n=1 infinity$(n<^>{-\alpha })_{n=1}<^>\infty$. For balls with an arbitrary sequence of radii, we find sharp bounds for the dimension and show that the natural extension of the Ekstr & ouml;m-Persson conjecture is not true in this case. Finally, we construct examples demonstrating that there does not exist a dimension formula involving only the lower and upper local dimensions of the measure and a critical parameter determined by the sequence of radii.