  The Andrews-Curtis conjecture claims that every normally generating $n$-tuple
of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of
Nielsen transformations and arbitrary conjugations. Replacing $F_n$ by an
arbitrary finitely generated group yields natural generalizations whose study
may help disprove the original and unsettled conjecture. We prove that every
finitely generated soluble group satisfies the generalized Andrews-Curtis
conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we
show that some soluble Baumslag-Solitar groups do not satisfy the generalized
Andrews-Curtis conjecture in the sense of Burns and Macedo\'nska.
