The polar layered deposits of Mars are ice-rich, finely stratified materials that may provide information on the climate history of the planet. The Mars Advanced Radar for Subsurface and Ionospheric Sounding (MARSIS) on the Mars Express orbiter obtained data over the North Polar Layered Deposits (NPLD) and found that they were easily penetrated to their base by the radar signal, and that the deposits must be nearly pure ice. This paper presents MARSIS observations of the South Polar Layered Deposits (SPLD) and surrounding terrain. MARSIS is a multi-frequency synthetic aperture orbital sounding radar. In its subsurface modes, MARSIS operates in four frequency bands between 1.3 and 5.5 MHz. The radar echoes from the surface of the SPLD are typically comparable in strength or weaker than those from surrounding smooth Mars surfaces. The signals clearly penetrate deep into the deposits at all frequency bands. In most cases, a strong reflection is seen at a time delay consistent with the expected depth of the contact of the SPLD materials with the substrate. Over the highest elevation SPLD surfaces, near 0° longitude, the estimated depth to the lower interface is over 3 km. In some areas, the lower interface is not clearly detected. Multiple internal reflectors are common. These can be quite continuous, over several 100 km in many cases. Some internal reflectors are substantially brighter than others, and some differences are seen in the apparent internal structure at different MARSIS frequencies. Thin (100s of m) layered units off the main SPLD are also penetrated by MARSIS, with a second interface detected at their lower boundaries. These units sometimes extend up to 1000 km northward from the SPLD. Their distribution roughly coincides with the unit mapped as the Hesperian aged "Dorsa Argentea" formation. Below the SPLD, the deep lower interfaces are interpreted as the base of the ice-rich unit, in contact with a relatively ice-poor substrate. The strong reflection from the lower interface implies very low loss and indicates that the bulk of the SPLD material is likely to be fairly pure ice, with dust fractions of probably only a few percent. The MARSIS data of the SPLD provide the opportunity to estimate the three-dimensional shape of the deposit. This allows for a new estimate of the water inventory of the SPLD, as well as further examination of effects of the overburden on the martian lithosphere.