Objectivity is the name given to covariance with respect to changes in observers by the action of the Euclidean group and is a central concept for mathematical modelling in Physics and Continuum mechanics. However, this notion is delicate and therefore hard to understand by both mechanicians and mathematicians, precisely because it resides at the junction of the two disciplines and hence appeals to knowledge and notions from both fields to become clear. In a first part ("behind objectivity") some classical notions of objectivity and frame-dependence are presented, discussed and critically revisited with the purpose to introduce in a second part ("beyond objectivity") a novel modelling approach in incompatible elasticity, based on intrinsic and geometric concepts, related but distinct from conventional objectivity. This contribution is primarily conceived to assist students, academics and researchers, mathematicians in particular, to find their way through some difficult and often ill-understood concepts of Physics and Continuum mechanics, since objectivity still remains nowadays a profound concept that deserves our attention.