  We consider ever-expanding Big Bang models with a cosmological constant,
$\Lambda$, and investigate in detail the evolution of the observable part of
the universe. We also discuss quintessence models from the same point of view.
  A new concept, the $\Lambda$-sphere (or $Q$-sphere, in the case of
quintessence) is introduced. This is the surface in our visible universe which
bounds the region where dark energy dominates the expansion, and within which
the universe is accelerating. We follow the evolution of this surface as the
universe expands, and we also investigate the evolution of the particle and
event horizons as well as the Hubble surface. We calculate the extent of the
observable universe and the portion of it that can be seen at different epochs.
Furthermore, we trace the changes in redshift, apparent magnitude and apparent
size of distant sources through cosmic history. Our approach is different from,
but complementary to, most other contemporary investigations, which concentrate
on the past light cone at the present epoch.
  When presenting numerical results we use the FRW world model with
$\Omega_{m0} = 0.30$ and $\Omega_{\Lambda0} = 0.70$ as our standard
cosmological model. In this model the $\Lambda$-sphere is at a redshift of
0.67, and within a few Hubble times the event horizon will be stationary at a
fixed proper distance of 5.1 Gpc (assuming $h_0 = 0.7$). All cosmological
sources with present redshift larger than 1.7 have by now crossed the event
horizon and are therefore completely out of causal contact.
