  C-loops are loops satisfying the identity $x(y\cdot yz) = (xy\cdot y)z$. We
develop the theory of extensions of C-loops, and characterize all nuclear
extensions provided the nucleus is an abelian group. C-loops with central
squares have very transparent extensions; they can be built from small blocks
arising from the underlying Steiner triple system. Using these extensions, we
decide for which abelian groups $K$ and Steiner loops $Q$ there is a
nonflexible C-loop $C$ with center $K$ such that $C/K$ is isomorphic to $Q$. We
discuss possible orders of associators in C-loops. Finally, we show that the
loops of signed basis elements in the standard real Cayley-Dickson algebras are
C-loops.
