  We discuss some of the issues which we encounter when we try to invoke the
scalar-tensor theories of gravitation as a theoretical basis of quintessence.
One of the advantages of appealing to these theories is that they allow us to
implement the scenario of a ``decaying cosmological constant,'' which offers a
reasonable understanding of why the observed upper bound of the cosmological
constant is smaller than the theoretically natural value by as much as 120
orders of magnitude. In this context, the scalar field can be a candidate of
quintessence in a broader sense. We find, however, a serious drawback in the
prototype Brans-Dicke model with $\Lambda$ added; a static universe in the
physical conformal frame which is chosen to have constant particle masses. We
propose a remedy by modifying the matter coupling of the scalar field taking
advantage of scale invariance and its breakdown through quantum anomaly. By
combining this with a conjecture on another cosmological constant problem
coming from the vacuum energy of matter fields, we expect a possible link
between quintessence and non-Newtonian gravity featuring violation of Weak
Equivalence Principle and intermediate force range, likely within the
experimental constraints. A new prediction is also offered on the
time-variability of the gravitational constant.
