  We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan
about the multiplicity of graded Cohen-Macaulay algebras over a field, for
codimension two algebras and for Gorenstein algebras of codimension three. In
fact, we prove stronger bounds than the conjectured ones allowing us to
characterize the extremal cases. This may be seen as a converse to the
multiplicity formula of Huneke and Miller that inspired the conjectural bounds.
