  We study the extraordinary dimension function dim_{L} introduced by
\v{S}\v{c}epin. An axiomatic characterization of this dimension function is
obtained. We also introduce inductive dimensions ind_{L} and Ind_{L} and prove
that for separable metrizable spaces all three coincide. Several results such
as characterization of dim_{L} in terms of partitions and in terms of mappings
into $n$-dimensional cubes are presented. We also prove the converse of the
Dranishnikov-Uspenskij theorem on dimension-raising maps.
