The high cross-section for the production of b quarks at the
Tevatron hadron collider make it an ideal place to search for rare B meson
species and very low branching fraction decay modes of the more common B
mesons.  This paper reports the recent discovery of the B_c meson by the CDF
collaboration.  It also discusses the observation of rare B meson final states
involving psi(2S) mesons and searches for ultra-rare decays such as B ->
mu^+mu^-.