Determining how the heliospheric magnetic field and plasma connect to the Sun's corona and photosphere is, perhaps, the central problem in solar and heliospheric physics. For much of the heliosphere, this connection appears to be well understood. It is now generally accepted that so-called coronal holes, which appear dark in X-rays and are predominantly unipolar at the photosphere, are the sources of quasi-steady wind that is generally fast, > 500 km/s, but can sometimes be slow. However, the connection to the Sun of the slow, non-steady wind is far from understood and remains a major mystery. We review the existing theories for the sources of the non-steady wind and demonstrate that they have difficulty accounting for both the observed composition of the wind and its large angular extent. A new theory is described in which this wind originates from the continuous opening and closing of narrow open field corridors in the corona, which give rise to a web of separatrices (the S-Web) in the heliosphere. Note that in this theory the corona-heliosphere connection is intrinsically dynamic, at least for this type of wind. Support for the S-Web model is derived from MHD solutions for the corona and wind during the time of the August 1, 2008 eclipse. Additionally, we perform fully dynamic numerical simulations of the corona and heliosphere in order to test the S-Web model as well as the interchange model proposed by Fisk and co-workers. We discuss the implications of our simulations for the competing theories and for understanding the corona-heliosphere connection, in general.