The aim of this article has been to recount how we go about describing two epochs of the Universe's evolution during which it may have experienced accelerated expansion, and to highlight possible similarities in those descriptions. It is an appealing possibility that the same underlying mechanism may be responsible in each case.
Concerning the early Universe, I have introduced some of the facets of inflation in a fairly simple manner. If you are interested in going beyond this, a detailed account of all aspects of the inflationary cosmology can be found in Ref. [3]. Additional material on particle physics and model building aspects of inflation can be found in Ref. [11].
At present, inflation is the most promising candidate theory for the origin of perturbations in the Universe. Different inflation models lead to discernibly different predictions for these perturbations, and hence high-accuracy measurements are able to distinguish between models, excluding either all or the vast majority of them. Since its inception, the inflationary cosmology has been a gallery of different models, and the gallery has continually needed extension after extension to house new acquisitions. In all the time up to the present, very few models have been discarded. However, the near future holds great promise to finally begin to throw out inferior models, and, if the inflationary cosmology survives as our model for the origin of structure, we can hope to be left with only a narrow range of models to choose between.
Concerning the present Universe, quintessence is an interesting idea which is still in its early days as far as observations are concerned. It provides a framework in which to study the possibility of an accelerating Universe at present, but so far it has not completely lived up to its initial promise as a way of challenging problems such as the coincidence problem. There appears to be plentiful scope for further investigations of possible quintessence scenarios to try and remedy this shortcoming, especially in anticipation of improved observations in years to come. At present, there is little in the way of rivals to quintessence in terms of allowing a quantitative description of the recent evolution of the Universe.