The core of a virus is simply the virion's genetic material (DNA or RNA) surrounded by a shell called the capsid, which is a very complex protein structure that is assembled by the host cell. Some more complex viruses also have an additional level of covering over the core called an envelope. The smallpox virus is one of the more famous examples of an extremely complex viral structure complete with a viral envelope. Outside of the entire virion is a covering called the surface membrane.