  C-loops are loops satisfying $x(y(yz))=((xy)y)z$. They often behave
analogously to Moufang loops and they are closely related to Steiner triple
systems and combinatorics. We initiate the study of C-loops by proving: (i)
Steiner loops are C-loops, (ii) C-loops are alternative, inverse property loops
with squares in the nucleus, (iii) the nucleus of a C-loop is a normal
subgroup, (iv) C-loops modulo their nucleus are Steiner loops, (v) C-loops are
power associative, power alternative but not necessarily diassociative, (vi)
torsion commutative C-loops are products of torsion abelian groups and torsion
commutative 2-C-loops; and several other results. We also give examples of the
smallest nonassociative C-loops, and explore the analogy between commutative
C-loops and commutative Moufang loops.
