A supergiant can be of varying colors. Most supergiants are either blue supergiants existing on the main sequence as type O or B and red supergiants existing off the main sequence. Examples of red supergiants include Antares and Betelgeuse and examples of blue supergiants include Rigel.
The modelling of supergiants is still an active area of research and is made more difficult by issues such as stellar mass loss. Rather than modelling individual stars, the latest trend has been to model clusters of stars and then compare the distribution of the resulting models with the observed supergiant distributions in things like the Magellanic Clouds.
The early universe is believed to have contained large number of supergiants known as population III stars. Their existence is necessary to explain observations of elements other than hydrogen and helium in quasars.
The consensus is that large stars move back and forth across the H-R diagrams during their lives. For some time, it was believed that large stars would become red supergiants near the end of their lives before going supernova. However, the progenitor for Supernova 1987A was a blue supergiant.