  One of brilliant achievements in spin-orbit-coupled Bose-Einstein condensates
is the discovery and observation of the supersolid stripe states. So far, all
studied supersolid stripe states do not carry supercurrent. In this work, we
reveal the existence of supercurrent-carrying supersolids in spin-orbit-coupled
Bose-Einstein condensates. The supersolid family has a parabolic-like
dispersion relation and carries supercurrent which is proportional to the
quasimomentum. Energetic and dynamical instabilities can break
supercurrent-carrying ability of this supersolid family. An insightful
interpretation of the dynamical instability of supercurrent-carrying
supersolids from the pure plane-wave phase is provided.
