  In this paper we develop a general framework of badly approximable points in
a metric space $X$ equipped with a $\sigma$-finite doubling Borel regular
measure $\mu$. We establish that under mild assumptions the $\mu$-measure of
the set of badly approximable points is always zero. The framework can be
applied to a variety of settings in Diophantine approximation and dynamical
systems, which we also consider, including weighted and $S$-arithmetic
Diophantine approximations, Diophantine approximation on manifolds and
intrinsic approximations on fractals.
