  We present a microscopic derivation of the defocusing two-dimensional cubic
nonlinear Schr\"odinger equation as a mean field equation starting from an
interacting $N$-particle system of Bosons. We consider the interaction
potential to be given either by $W_\beta(x)=N^{-1+2 \beta}W(N^\beta x)$, for
any $\beta>0$, or to be given by $V_N(x)=e^{2N} V(e^N x)$, for some spherical
symmetric, positive and compactly supported $W,V \in
L^\infty(\mathbb{R}^2,\mathbb{R})$. In both cases we prove the convergence of
the reduced density matrix corresponding to the exact time evolution to the
projector onto the solution of the corresponding nonlinear Schr\"odinger
equation in trace norm. For the latter potential $V_N$ we show that it is
crucial to take the microscopic structure of the condensate into account in
order to obtain the correct dynamics.
