  Over a Cohen-Macaulay local ring, the minimal number of generators of a
maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984
Ulrich asked whether there always exist modules for which equality holds; such
modules are known nowadays as Ulrich modules. We answer this question in the
negative by constructing families of two dimensional Cohen-Macaulay local rings
that have no Ulrich modules. Some of these examples are Gorenstein normal
domains; others are even complete intersection domains, though not normal.
