The N-extended massless superparticle in a 4D Minkowski background has a worldline time-reversal invariance that becomes CPT in the quantum theory. This symmetry is anomalous for odd N, which explains why there is no CPT-self-conjugate massless supermultiplet for N=1. In the supertwistor formulation of superparticle mechanics, the anomaly is a close cousin of the parity anomaly of 3D gauge theories. CPT is not anomalous if N is even but if 2N is odd there is a Kramers degeneracy, which explains why the N=2 hypermultiplet is a doubled version of the CPT-self-conjugate supermultiplet of helicities (-1/2,0,0,1/2).

Non-zero mass is compatible with unbroken gauge invariance in three spacetime dimensions (3D). A systematic procedure for the construction of massive gauge theories will be illustrated by new massive gravity, which propagates unitarily two massive spin 2 modes in a Minkowski vacuum. The supergravity extension of this model will be presented along with new results on supersymmetric AdS vacua. The extension to a new N=8 3D supergravity will be discussed, as will be the AdS3CFT2 correspondence and possible connections to string/M-theory.