A wave map is a map \(\varphi\) from Minkowski spacetime \(\mathbb R^{n+1}\) to a target Riemannian manifold \((M,g)\) which is a critical point of the functional \[ \int\int \langle\partial_\alpha \varphi,\partial^\alpha \varphi\rangle_g dx dt. \] The Euler-Lagrange equation for this function in local coordinates takes the form \[ \square\varphi= -\Gamma(\varphi)(\partial_\alpha \varphi,\partial^\alpha\varphi). \] In this work the author obtains global well-posedness and regularity for the wave maps equation in all dimensions \(n\geq 2\) provided that the initial data \((\varphi(0),\varphi_t(0))\) is small in the homogeneous Besov norm \(\dot B^{n/2,1}_2\times\dot B^{n/2-1,1}_2\), which is essentially the same as the critical Sobolev norm \(\dot H^{n/2}\times\dot H^{n/2-1}\) but is a slightly larger norm because the Littlewood-Paley components of the data are summed in \(l^1\) rather than \(l^2\). This is essentially the best possible small-data regularity result one can obtain by purely iterative methods, as this norm is essentially the last norm which can control \(L^\infty\). The result is obtained by an iteration argument in a rather complicated space, which is at heart based on the \(X^{s,b}\) spaces of Bourgain and Klainerman-Machedon, but with some necessary (and rather ingenious) modifications to deal with the critical nature of the problem, and specifically the fact that \(X^{n/2,1/2}\) fails by a logarithm to control \(L^\infty\). The same type of spaces have been used in later work to obtain global regularity for the critical Sobolev spaces themselves.