INFLATION WITH OMEGA-NOT-EQUAL-1

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