  A finitely generated quadratic module or preordering in the real polynomial
ring is called stable, if it admits a certain degree bound on the sums of
squares in the representation of polynomials. Stability, first defined
explicitly by Powers and Scheiderer, is a very useful property. It often
implies that the quadratic module is closed; furthermore it helps settling the
Moment Problem, solves the Membership Problem for quadratic modules and allows
applications of methods from optimization to represent nonnegative polynomials.
  We provide sufficient conditions for finitely generated quadratic modules in
real polynomial rings of several variables to be stable. These conditions can
be checked easily. For a certain class of semi-algebraic sets, we obtain that
the nonexistence of bounded polynomials implies stability of every
corresponding quadratic module. As stability often implies the non-solvability
of the Moment Problem, this complements Schmuedgen's result which uses bounded
polynomials to check the solvability of the Moment Problem by dimensional
induction.
  We also use stability to generalize a result on the Invariant Moment Problem
by Cimpric, Kuhlmann and Scheiderer.
