In the advanced course given at Centre de Recerca Matem`atica, consisting of twelve hour lectures from 22 February to 5 March 2010, we described results and discussed some open problems regarding the gamma and zeta functions in the function field context. The first four lectures of these notes, dealing with gamma, roughly correspond to the first four lectures of one and half hour each, and the last three lectures, dealing with zeta, cover the last three two-hour lectures. Typically, in each part, we first discuss elementary techniques, then easier motivating examples with Drinfeld modules in detail, and then outline general results with higher-dimensional t-motives. Lecture 4 is independent of Lecture 3, whereas the last part (last three lectures) is mostly independent of the first part, except that the last two lectures depend on Lecture 3. At the end, we include a guide to the relevant literature.