  Testing procedures for assessing specific parametric model forms, or for
checking the plausibility of simplifying assumptions, play a central role in
the mathematical treatment of the uncertain. No certain answers are obtained by
testing methods, but at least the uncertainty of these answers is properly
quantified. This is the case for tests designed on the two most general data
generating mechanisms in practice: distribution/density and regression models.
Testing proposals are usually formulated on the Euclidean space, but important
challenges arise in non-Euclidean settings, such as when directional variables
(i.e., random vectors on the hypersphere) are involved. This work reviews some
of the smoothing-based testing procedures for density and regression models
that comprise directional variables. The asymptotic distributions of the
revised proposals are presented, jointly with some numerical illustrations
justifying the need of employing resampling mechanisms for effective test
calibration.
