Arakelov geometry and the Voevodsky-style theory of algebraic homology or homotopy (variants of which were discovered and studied by Suslin, Rost, Levine, Morel, Hanamura and Voevodsky, to quote just a few) provide entirely new methods of investigation for problems that have resisted the efforts of many researchers. In this project, our aim is twofold: 1> First of all, we intend to use the objects and the tools of the theory of scheme-theoretic homotopy to solve certain difficult problems coming from Arakelov theory. In particular, it seems to us that even to formulate, let alone prove the most general fixed point theorems (as well as their arakelov-theoretic extensions), this new framework is required. We thus hope to get closer to the proof of some very general conjectures that we made a few years ago. These conjectures have important consequences for the arithmetic geometry of Shimura varieties. 2> Reciprocally, we want to extend the constructions of Arakelov theory to the much more general context of scheme-theoretic homotopy. We expect that two types of results will be reached, if we carry out that plan: - New and very general constructions of non-trivial elements in motivic cohomology (please note that we have already obtained a few partial but original results in this direction), in the context of Beilinson's conjectures and its variants. - Results of finite generation, in some cases, for the motivic cohomology groups, thus generalising the classical approach of Dirichlet.