I'm looking for a reference that gives an overview of the most important properties of Arakelov intersection theory (on arithmetic varieties of arbitrary dimension) and that describes basic properties of the Arakelov Chow ring. There's a similar MO question asking about survey articles on (classical) intersection theory, so I guess that I'm asking the same question, but for Arakelov intersection theory.

In addition to A. Javanpeykar's references, you might want to look at the paper "Arithmetic intersection theory" by H. Gillet and C. Soulé (Pub. Math. de l'IHES, tome 72, no. 2 (1990)). – Damian Rössler Jul 28 '12 at 12:24
A good reference in my humble opinion is Bost's paper in Bourbaki:
Théorie de l'intersection et théorème de RiemannRoch arithmétiques
Séminaire BOURBAKI. Novembre 1990. 43ème année, 199091, n° 731
Another reference would be Soule's book on Arakelov geometry "Lectures on Arakelov geometry" written with Abramovich, Burnol and Kramer.
Finally, I know you didn't ask this, but in the case of arithmetic surfaces there are more references (besides Faltings' "Calculus on arithmetic surfaces" and Arakelov's original paper). For example, Deligne's paper "le determinant de la cohomologie" and R. de Jong's Ph.D. thesis: http://www.math.leidenuniv.nl/~rdejong/publications/thesis.pdf Also, MoretBailly's paper "Metriques permises" in Szpiro's 1985 Asterisque is wonderful.

@Ariyan: Thanks, I'll take a look at those. I'm okay (moreorless) with arithmetic surfaces. In that case, in addition to the references you mention, there's also Lang's book. – Joe Silverman Jul 27 '12 at 15:22

2@Joe. You're welcome. I was just about to edit my answer and add Lang's book which is of course also a great source. Also, in Bost's paper, the basic properties of the intersection product are summarized in Theoreme 2.5 page 59. Also, I think the properties of arithmetic characteristic classes are all in Theoreme 3.2 and Theoreme 3.3. The rest of the article (Section 4) is devoted to the arithmetic RiemannRoch theorem in degree 1. The arithmetic RiemannRoch theorem in higher degree is proven in a recent paper by Gillet, Rossler and Soule. aif.cedram.org/item?id=AIF_2008__58_6_2169_0 – Ariyan Javanpeykar Jul 27 '12 at 15:37

2Bost's talk is available at Numdam: numdam.org/numdambin/fitem?id=SB_19901991__33__43_0 – Chandan Singh Dalawat Aug 1 '12 at 14:46