We describe a Sun-to-Earth system of coupled models. Our main goal is to create a real-time, three-dimensional (3-D), MHD-based system to aid in the operational forecasting of geomagnetic activity, but we expect the system to have other uses. We give here our initial survey of the system's characteristics. The Hybrid Heliospheric Modeling System (HHMS) is composed of two physics-based models, combined with two simple empirical models. The physics-based models are a source surface (potential field) current sheet model for the corona and a time-dependent 3-D MHD solar wind model. The system is driven by a sequence of photospheric magnetic maps composed from daily magnetograms. An empirical relationship between magnetic flux tube expansion factor and solar wind speed at 0.1 AU is a key element of the system. The solar wind model gives a predicted time series of MHD parameters at the location of Earth in the model grid; this is verified against Omni, Wind, or ACE satellite data, depending on the time period. The predicted solar wind at Earth is used as input to the second, data-based, empirical model to predict the geomagnetic Ap index. We compare test results for simulated 1 day ahead Ap forecasts for the years 1993 through 2002 with forecast skill of the official Ap forecasts that were issued by the NOAA Space Environment Center in that time interval. Results show the HHMS would have been useful to forecasters in some years. Simulations of transient events such as coronal mass ejections and interplanetary shocks with the HHMS will be reported on later.