Let X be a 2-dimensional, normal, flat, proper scheme over the integers. Assume ¯L and ¯M are two hermitian line bundles over X. Arakelov (and Deligne) defined a real number ¯L.¯M, the arithmetic intersection number of ¯L and ¯M. We shall explain the definition and the basic properties of this number. Next, we shall see how to extend this construction to higher dimension, and how to interpret it in terms of arithmetic Chow groups.