  We examine existence and uniqueness of strong solutions of multi-dimensional
mean-field stochastic differential equations with irregular drift coefficients.
Furthermore, we establish Malliavin differentiability of the solution and show
regularity properties such as Sobolev differentiability in the initial data as
well as H\"older continuity in time and the initial data. Using the Malliavin
and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula
for mean-field stochastic differential equations, i.e. a probabilistic
representation of the first order derivative of an expectation functional with
respect to the initial condition.
