  In this paper we study the regularity of stationary and minimizing harmonic
maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If
$S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$
is $k^{th}$-stratum of the singular set of $f$, then it is well known that
$\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is
understood in any generality. Our first result is for a general stationary
harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable.
  In the case of minimizing harmonic maps we go further, and prove that the
singular set $S(f)$, which is well known to satisfy $\dim S(f)\leq n-3$, is in
fact $n-3$-rectifiable with uniformly {\it finite} $n-3$-measure. An effective
version of this allows us to prove that $|\nabla f|$ has estimates in
$L^3_{weak}$, an estimate which is sharp as $|\nabla f|$ may not live in $L^3$.
  The above results are in fact just applications of a new class of estimates
we prove on the {\it quantitative} stratifications $S^k_{\epsilon,r}(f)$ and
$S^k_{\epsilon}(f)\equiv S^k_{\epsilon,0}(f)$. Roughly,
$S^k_{\epsilon}\subseteq M$ is the collection of points $x\in S^k_\epsilon$ for
which no ball $B_r(x)$ is $\epsilon$-close to being $k+1$-symmetric. We show
that $S^k_\epsilon$ is $k$-rectifiable and satisfies the Minkowski estimate
$Vol(B_r\,S_\epsilon^k)\leq C r^{n-k}$.
  The proofs require a new $L^2$-subspace approximation theorem for stationary
harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg
type theorems. These results are generalizations of the classical Reifenberg,
and give checkable criteria to determine when a set is $k$-rectifiable with
uniform measure estimates. The new Reifenberg type theorems may be of some
independent interest.
