  We take advantage of the correspondence between pseudogroups and inverse
quantal frames, and of the recent description of Morita equivalence for inverse
quantal frames in terms of biprincipal bisheaves, to define Morita equivalence
for pseudogroups and to investigate its applications. In particular, two
pseudogroups are Morita equivalent if and only if their corresponding localic
\'etale groupoids are. We explore the clear analogies between our definition of
Morita equivalence for pseudogroups and the usual notion of strong Morita
equivalence for $C^{\ast}$-algebras and these lead to a number of concrete
results.
