We analyze the transverse-momentum-dependent distribution and fragmentation functions in space-like and time-like hard processes involving at least two hadrons, in particular, 1-particle inclusive leptoproduction, the Drell-Yan process and two-particle inclusive hadron production in electron-positron annihilation. As is well known, transverse momentum dependence allows for the appearance of unsuppressed single spin azimuthal asymmetries, such as Sivers and Collins asymmetries. Recently, Belitsky, Ji and Yuan obtained fully color-gauge-invariant expressions for the relevant matrix elements appearing in these asymmetries at leading order in an expansion in the inverse hard scale. We rederive these results and extend them to observables at the next order in this expansion. We observe that at leading order one retains a probability interpretation, contrary to a claim in the literature and show the direct relation between the Sivers effect in single spin asymmetries and the Qiu-Sterman mechanism. We also study fragmentation functions, where the process-dependent gauge link structure of the correlators is not the only source of T-odd observables and discuss the implications for universality. © 2003 Elsevier B.V. All rights reserved.