  For any $n$-dimensional compact spin Riemannian manifold $M$ with a given
spin structure and a spinor bundle $\Sigma M$, and any compact Riemannian
manifold $N$, we show an $\epsilon$-regularity theorem for weakly
Dirac-harmonic maps . As a consequence, any weakly Dirac-harmonic map is proven
to be smooth when n = 2. A weak convergence theorem for approximate
Dirac-harmonic maps is established when $n = 2$. For $n \ge 3$, we introduce
the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem
for stationary Dirac-harmonic maps in $R^n$. If, additions, $\psi\in W^{1,p}$
for some $p>2n/3$, then we obtain an energy monotonicity formula and prove a
partial regularity theorem for any such a stationary Dirac-harmonic map.
