  Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which
supports a non-atomic finite measure m. We consider `natural' classes of badly
approximable subsets of \Omega. Loosely speaking, these consist of points in
\Omega which `stay clear' of some given set of points in X. The classical set
\Bad of `badly approximable' numbers in the theory of Diophantine approximation
falls within our framework as do the sets \Bad(i,j) of simultaneously badly
approximable numbers. Under various natural conditions we prove that the badly
approximable subsets of \Omega have full Hausdorff dimension. Applications of
our general framework include those from number theory (classical, complex,
p-adic and formal power series) and dynamical systems (iterated function
schemes, rational maps and Kleinian groups).
