We investigate the relations between the cut number, c(M ), and the first Betti number, b 1 (M ), of 3-manifolds M. We prove that the cut number of a "generic" 3-manifold M is at most 2. This is a rather unexpected result since specific examples of 3-manifolds with large b 1 (M ) and c(M ) ≤ 2 are hard to construct. We also prove that for any complex semisimple Lie algebra g there exists a 3-manifold M with b 1 (M ) = dim g and c(M ) ≤ rank g. Such manifolds can be explicitly constructed.