  The purpose of this article is to initiate Arakelov theory in a
noncommutative setting. More precisely, we are concerned with Arakelov theory
of noncommutative arithmetic curves. Our first main result is an arithmetic
Riemann-Roch formula in this setup. We proceed with introducing the
Grothendieck group of arithmetic vector bundles on a noncommutative arithmetic
curve and show that there is a uniquely determined degree map, which we then
use to define a height function. We prove a duality theorem for this height.
