  In this paper, we first establish well-posedness of McKean-Vlasov stochastic
differential equations (McKean-Vlasov SDEs) with common noise, possibly with
coefficients having super-linear growth in the state variable. Second, we
present stable time-stepping schemes for this class of McKean-Vlasov SDEs.
Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for
an interacting particle system associated with the McKean-Vlasov equation. We
prove stability and strong convergence of order $1/2$ and $1$, respectively. To
obtain our main results, we employ techniques from calculus on the Wasserstein
space. The proof for the strong convergence of the tamed Milstein scheme only
requires the coefficients to be once continuously differentiable in the state
and measure component. To demonstrate our theoretical findings, we present
several numerical examples, including mean-field versions of the stochastic
$3/2$ volatility model and the stochastic double well dynamics with
multiplicative noise.
