Many versions of Tate conjectures were proved for Drinfeld modules and for their higher-dimensional generalizations, the t-modules of Anderson. The underpinning of this success is a technically simple but powerful theory of t-motives pioneered by Anderson. In this talk I shall describe an approach to Tate conjectures for t-modules which implies all the known versions and explains why some variants of the conjectures fail. The approach combines the theory of t-motives with the t-adic counterpart of the theory of overconvergent F-isocrystals.