is essentially the only situation in building theory we consider in this paper*]{}. Its particularly nice features are, among others, the facts that in this case the buildings are (contractible, hence simply connected) gluings of Euclidean tilings and that deep (non-positive curvature) metric arguments are therefore available; moreover, on the group side, structures are shown to be even richer than expected. For instance, topologically the action on the buildings enables one to classify and understand maximal compact subgroups (which is useful to representation theory and harmonic analysis) and, algebraically, it enables one to define important integral models for the group (which is again useful to representation theory, and which is also a crucial step towards analytic geometry).

One delicate point in this theory is merely to prove that for a suitable non-Archimedean reductive group, there does exist a nice action on a suitable Euclidean building: this is the main achievement of the work by F. Bruhat and J. Tits in the 70ies [@BT1a], [@BT1b]. Eventually, Bruhat-Tits theory suggests to see the Euclidean buildings attached to reductive groups over valued fields (henceforth called [*Bruhat-Tits buildings*]{}) as non-Archimedean analogues of the symmetric spaces arising from real reductive Lie groups, from many viewpoints 