As epsilon-constants show up in (local or global) functional equations, they also play a crucial role regarding  the behaviour of the (Equivariant) Tamagawa Number Conjecture when going over from a motive M to its (Kummer) dual M*(1). The existence of epsilon-isomorphisms - in the language of Fukaya & Kato - or the Local Tamagawa Number Conjecture - in the terminology of Burns & Flach (generalizing Fontaine & Perrin-Riou) - just imply the compatibility of the global conjecture for M and M*(1) taking into account  their  functional equation and Poitou-Tate/Artin-Verdier duality, respectively. In the talk we shall report on recent improvements and results in this direction, which is partly joint work with David Loeffler and Sarah Zerbes.