In this paper we study Krull domains graded by an arbitrary torsionless grading monoid г. We generalize many of the well-known results for Krull domains graded by the integers. If г⊂(-г)=0, then there is an injection C1(A0)→C1(A), which need not be a homorphism. We show that C1(A) is generated by the classes of the homogeneous height-one prime ideals of A. Probably the most natural г-graded domain is A[г]. Let A be a Krull domain with quotient field K. If A[г] is a Krull domain, then C1(A[г])=C1(A)⊕C1(K[г]) and C1(K[г]) is independent of K. If г is finitely generated, we give necessary and sufficient conditions or г for A[г] to be a Krull domain. In this case A[г] is just a subring of somegenerated by monomials. We also study graded domains in which all nonzero homogeneous elements are units. Such graded domains are very close to being group rings over a field.