  The spinor representation is developed and used to investigate minimal
surfaces in ${\bfR}^3$ with embedded planar ends. The moduli spaces of
planar-ended minimal spheres and real projective planes are determined, and new
families of minimal tori and Klein bottles are given. These surfaces compactify
in $S^3$ to yield surfaces critical for the M\"obius invariant squared mean
curvature functional $W$. On the other hand, all $W\!$-critical spheres and
real projective planes arise this way. Thus we determine at the same time the
moduli spaces of $W\!$-critical spheres and real projective planes via the
spinor representation.
