On the Cosmological Constant Problems and the Astronomical Evidence for a Homogeneous Energy Density with Negative Pressure

In this article the cosmological constant problems, as well as the astronomical evidence for a cosmologically significant homogeneous exotic energy density with negative pressure (quintessence), are reviewed for a broad audience of physicists and mathematicians. After a short history of the cosmological term it is explained why the (effective) cosmological constant is expected to obtain contributions from short-distance physics corresponding to an energy scale at least as large as the Fermi scale. The actual tiny value of the cosmological constant by particle physics standards represents, therefore, one of the deepest mysteries of present-day fundamental physics. In a second part I shall discuss recent astronomical evidence for a cosmologically significant vacuum energy density or an effective equivalent, called quintessence. Cosmological models, which attempt to avoid the disturbing cosmic coincidence problem, are also briefly reviewed.