The first Teraflop/s computer, the ASCI Red, became operational in 1997, and it took more than 11 years for a Petaflop/s performance machine, the IBM Roadrunner, to appear on the Top500 list. Efforts have begun to study the hardware and software challenges for building an exascale machine. It is important to understand and meet these challenges in order to attain Exaflop/s performance. This paper presents a feasibility study of three important application classes to formulate the constraints that these classes will impose on the machine architecture for achieving a sustained performance of 1 Exaflop/s. The application classes being considered in this paper are - classical molecular dynamics, cosmological simulations and unstructured grid computations (finite element solvers). We analyze the problem sizes required for representative algorithms in each class to achieve 1 Exaflop/s and the hardware requirements in terms of the network and memory. Based on the analysis for achieving an Exaflop/s, we also discuss the performance of these algorithms for much smaller problem sizes.