  We present a space-time least squares finite element method for the heat
equation. It is based on residual minimization in L2 norms in space-time of an
equivalent first order system. This implies that (i) the resulting bilinear
form is symmetric and coercive and hence any conforming discretization is
uniformly stable, (ii) stiffness matrices are symmetric, positive definite, and
sparse, (iii) we have a local a-posteriori error estimator for free. In
particular, our approach features full space-time adaptivity. We also present
a-priori error analysis on simplicial space-time meshes which are highly
structured. Numerical results conclude this work.
