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 $ϕ= 0$ is unstable but the effective potential has a discrete symmetry $ϕ\to =-ϕ$. 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.

21 pages, 17 figures (not included), Stanford University preprint SU--ITP--94--3