  The aim of this note is to exhibit explicit sufficient criteria ensuring
bigness of globally generated, rank-$r$ vector bundles, $r \geqslant 2$, on
smooth, projective varieties of even dimension $d \leqslant 4$. We also discuss
connections of our general criteria to some recent results of other authors, as
well as applications to tangent bundles of Fano varieties, to suitable
Lazarsfeld-Mukai bundles on four-folds, etcetera.
