The theory of quasiconformal mappings generalizes to higher dimensions the geometric viewpoint in complex analysis. Foundations for the subject were laid by F. W. Gehring and the Finnish school (and independently by the Soviets), who established basic characterizations and analytic properties of quasiconformal mappings in Euclidean spaces of dimension at least two. Recent developments in geometry have highlighted the need for abstract formulations of this theory. We modify Pansu's generalized notion of the modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension Q > 1, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three different definitions for quasiconformality: (1) an infinitesimal metric condition: arbitrarily small annuli centered at any point in the domain are taken to configurations of bounded modulus in the target; (2) a semi-global metric condition: on each compact subset of the domain, the map distorts the relative distance between triples of points by a bounded factor; (3) a potential-theoretic characterization: the ratio of the conformal modulus of any collection of curves to that of its image is uniformly bounded away from zero and infinity. Moreover, we prove that such maps are absolutely continuous in measure if and only if they are absolutely continuous along Q -almost every curve; we conjecture that either one (and hence both) of these results hold in this setting. We illustrate our results with several corollaries. First, we show that any Q-regular Loewner space, Q > 1, for which closed balls are compact, is not quasiconformally equivalent to any proper subdomain. In the Euclidean case this result is due to Loewner. Second, for each Q ≥ 1, we exhibit compact Q-regular spaces all of whose quasisymmetric images have dimension at least Q. We conjecture that any space with dimension Q < 1 can always be quasisymmetrically deformed onto spaces of arbitrarily small dimension. Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent. Here, for 0 < e≤1 , we define the snowflake curve of order e to be the real line R with the metric dx,y=x-y e.