Can planets exist in a binary or multiple system? Have scientists ever discovered such a solar system?
Planets can exist in binary systems, but not on any kind of orbit and not in any kind of a binary system. Few extrasolar planets have been found around a star that is a member of a binary, and each time the planet was much closer to its parent star that the distance between the two stars. Any planets that would sometimes get too close to the other sun are expected to have unstable orbits that would eventually put them on collision course with one of the stars. It should be noted that, while most binaries in which planets have been found had exceptionally large separation between stars (like in 16 Cygni and Upsilon Andromedae systems), this might be due to an observational bias. Usually, "planet hunters" go either for single stars or very wide binaries, where there is little interference from the companion, since planets are easier to detect in those cases.
The discovery of the first planet in a relativly close binary system was made by William Cochran and his team (at McDonald Observatory in Texas) and announced just this month. Here is the link (you can read their press release here). Note that while this star (Gamma Cephei) is not very close to Earth, it is very similar to Alpha Centauri system, the closest star to Earth, and this implies at least theoretical possibility that Alpha Centauri too might have planets.