The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper Imbedding of an abstract variety in a complete variety. Unfortunately, this paper was apparently written before the language of schemes became standard, and uses Nagata's own language for "algebraic geometry over a Dedekind domain." Does anyone know of a translation of this proof (or another of the same statement) into scheme-theoretic language (or other language more comprehensible to the contemporary reader)?
Apart from Brian's, published as:
Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257.
Lütkebohmert, On compactification of schemes. Manuscripta Math. 80 (1993), no. 1, 95–111.
Vojta: Nagata's embedding theorem, arXiv:0706.1907
Deligne: Le théorème de plongement de Nagata, Kyoto J. Math. 50, Number 4 (2010), 661-670.
All of them are worth reading. The issue is certainly subtle and important, at least for cohomological constructions.
Brian Conrad has a writeup on this: