  We investigate the global structure of inflationary universe both by
analytical methods and by computer simulations of stochastic processes in the
early Universe. We show that the global structure of the universe depends
crucially on the mechanism of inflation. In the simplest models of chaotic
inflation the Universe looks like a sea of thermalized phase surrounding
permanently self-reproducing inflationary domains. In the theories where
inflation occurs near a local extremum of the effective potential corresponding
to a metastable state, the Universe looks like de Sitter space surrounding
islands of thermalized phase. A similar picture appears even if the state $\phi
= 0$ is unstable but the effective potential has a discrete symmetry $\phi \to
=-\phi$. In this case the Universe becomes divided into domains containing
different phases. These domains will be separated from each other by domain
walls. However, unlike ordinary domain walls, these domain walls will inflate,
and their thickness will exponentially grow. In the theories with continuous
symmetries inflation generates exponentially expanding strings and monopoles
surrounded by thermalized phase. Inflating topological defects will be stable,
and they will unceasingly produce new inflating topological defects. This means
that topological defects may play a role of indestructible seeds for eternal
inflation.
