This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesnt require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized
rings and schemes, which include classical rings and schemes together with exotic objects such as F_1 (field with one element), Z_infty (real integers), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.
