A revolutionary change in cosmology has been underway for the last fifteen years -- the transition to precision cosmology. It is now common to discuss cosmological measurements at the percent level of accuracy or better -- in certain cases, achievable target accuracies as low as 0.1%! A powerful and diverse suite of cosmological observations has led to this remarkable transformation. Taken together, results from cosmic microwave background observations, studies of the large-scale distribution of matter in the Universe, and of the redshift distribution of supernovae, yield an impressively consistent picture of the fundamental make-up of the Universe, the so-called Concordance Model which requires the existence of a mysterious Dark Energy. The establishment of the concordance model has in turn ushered in the era of precision cosmology. Forecasts for determination of cosmological parameters from next-generation observations require calibration against large-scale cosmological simulations accurate to the level of one percent or better. The petabyte scale of simulation and observation datasets raises yet another set of challenges including unprecedented levels of automation. In this talk I will discuss where we stand today with regard to these challenges for next-generation computational cosmology. As a primary application, I will focus on how progress in high performance computing will impact dark energy studies, the primary target of future cosmological surveys.