Abstract: A key player in the Langlands program is the L-group ^LG of a connected reductive group G over a local or global field. It underpins both pillars of the program: the reciprocity conjecture, which relates automorphic representations of G to Galois representations valued in ^LG, and the functoriality conjecture, which relates automorphic representations of two different groups whose L-groups are related. We will discuss how considerations of functoriality lead to an extension of the concept of L-group to certain non-linear covering groups, and how these can be organized using a new kind of fundamental group associated to G. We will focus on the case of a local field.