Scientific evidence strongly indicates that human activities are playing a significant role in bringing about major changes in our climate system, particularly in the last five decades. Significant changes in climate as a result of these human activities are projected for the rest of the 21st Century and beyond. The scientific analyses also strongly indicate that the globally-averaged temperature charges are being largely driven by changing atmospheric concentration of radiatively important gases and particles, lending credence to concerns about much larger changes in temperature, precipitation, and for changing probability in the likelihood for severe weather events over the coming decades. Numerical models of the physics, chemistry, and biology affecting the Earth-atmosphere climate system are key tools to both understanding the past changes in climate and to making the projections for the future that are necessary for planning and policy purposes. Today, even as we prepare to run these complex models of the Earth's climate system on petascale machines, we realize that despite the extensive capabilities that petascale will enable, a number of critical limitations in modeling the climate system require an exascale capability. The purpose of this presentation is to discuss the science underlying the concerns about climate change, the models of the Earth's climate system used in studying climate change, and the challenges being faced in taking these models to petascale and exascale. The NCAR Community Earth System Model (CESM), the most cited and used climate model in the world, will be the basis for the discussion on climate modeling capabilities now and into the future.