  We review what is currently known about the transverse spin structure of
hadrons, in particular from observables that can be analyzed within a collinear
framework. These effects have been around for 40 years and represent a critical
test of perturbative QCD. We look at both proton-proton and lepton-nucleon
collisions for various final states. While the main focus is on transverse
single-spin asymmetries, we also discuss how longitudinal-transverse
double-spin asymmetries offer a complimentary, yet equally important, source of
information on the quark-gluon content of hadrons. We also summarize some
recent progress in solidifying the theoretical formalism behind these
observables and give an outlook on future directions of research.
