During the past two decades, convective scale models have advanced sufficiently to study the dynamic and microphysical processes associated with mesoscale convective systems. The basic features of these models are that they are non-hydrostatic and include a good representation of microphysical processes. The Goddard Cumulus Ensemble (GCE) model has been extensively applied to study cloud-environment interactions, cloud interaction and mergers, air-sea interaction, cloud draft structure and trace gas transport. The GCE model has improved significantly during the past decade. For example, ice-microphysical processes, and solar and infrared radiative transfer processes have been included. These model improvements allow the GCE model to study cloud-radiation interaction, cloud-radiation-climate relations and to develop rain retrieval algorithms for Tropical Rainfall Measuring Mission (TRMM). In Part I, a full description of the GCE model is presented, as well as several sensitivity tests associated with its assumptions. In Part II (Simpson and Tao, 1993), we will review GCE model applications to cloud precipitating processes and to the Tropical Rainfall Measuring Mission (TRMM), a joint U.S.-Japan satellite project to measure rain and latent heat release over the global tropics.