In algebraic number theory, a number field extension L/K is called tamely ramified at a prime ideal P if the ramification index r(Q/P) is relatively prime to the characteristic of the residue field for all prime ideals Q lying above P in the ring of integers O_L. The field extension itself is tamely ramified if it is tamely ramified at all primes P, and this notion of tameness is stable under composition of fields. As this makes it possible to consider the maximal tamely ramified field extension of a given number field K. The purpose of my talks is to investigate and compare analogous concepts in arithmetic algebraic geometry: Given a cover of arithmetic schemes Y --> X, there are several possible generalisations of the concept of tameness of field extensions, with different functoriality properties. Following and expanding on ideas introduced by G. Wiesend, M. Kerz and F. Pop, I shall present some of the properties of the resulting tame fundamental groups and show how they are related.