  We study normal analytic compactifications of C^2 and describe their
singularities and configuration of curves at infinity, in particular improving
and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by
product we give new proofs of Jung's theorem on polynomial automorphisms of C^2
and Remmert and Van de Ven's result that CP^2 is the only smooth analytic
compactification of C^2 for which the curve at infinity is irreducible. We also
give a complete answer to the question of existence of compactifications of C^2
with prescribed divisorial valuations at infinity. In particular, we show that
a valuation on C(x,y) centered at infinity determines a compactification of C^2
iff it is "positively skewed" in the sense of (Favre and Jonsson, Ann. Sci.
Ecole Norm. Sup. 40(2):309--349, 2007).
