  We prove the equivalence between geometric and analytic definitions of
quasiconformality for a homeomorphism $f\colon X\rightarrow Y$ between
arbitrary locally finite separable metric measure spaces, assuming no metric
hypotheses on either space. When $X$ and $Y$ have locally $Q$-bounded geometry
and $Y$ is contained in an Alexandrov space of curvature bounded above, the
sharpness of our results implies that, as in the classical case, the modular
and pointwise outer dilatations of $\map$ are related by $K_O(f)=
\operatorname{esssup} H_O(x,f)$.
