Inflation with Omega not equal 1

We discuss various models of an inflationary universe with Omega not equal 1. A homogeneous universe with Omega > 1 may appear due to the creation of a universe ''from nothing'' in theories where the effective potential becomes very steep at large phi, or in theories where the inflaton field phi nonminimally couples to gravity. Inflation with Omega < 1 generally requires an intermediate first-order phase transition with bubble formation and with a second stage of inflation inside the bubble. It is possible to realize this scenario in the context of a theory of one scalar field, but typically it requires artificially bent effective potentials and/or nonminimal kinetic terms. It is much easier to obtain an open universe in models involving two scalar fields. However, these models have their own specific problems. We propose three different models of this type which can describe an open homogeneous inflationary universe.