  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.
