  In this work we show that the classical subject of general valuation theory
and Zariski-Riemann varieties has a much wider scope than commutative algebra
and desingularization theory. We construct and investigate birational
projective limit objects appropriate for the study of countably many birational
models at one time. We use nonseparated Riemann varieties to investigate the
birational structure of integral preschemes satisfying the existence condition
of the valuative criterion of properness.
