  We perform a comprehensive study of the dominant two- and higher-loop
contributions to the $^{205}$Tl, neutron and muon electric dipole moments
induced by Higgs bosons, third-generation quarks and squarks, charginos and
gluinos in the Minimal Supersymmetric Standard Model (MSSM). We find that
strong correlations exist among the contributing CP-violating operators, for
large stop, gluino and chargino phases, and for a wide range of values of
$\tan\beta$ and charged Higgs-boson masses, giving rise to large suppressions
of the $^{205}$Tl and neutron electric dipole moments below their present
experimental limits. Based on this observation, we discuss the constraints that
the non-observation of electric dipole moments imposes on the
radiatively-generated CP-violating Higgs sector and on the mechanism of
electroweak baryogenesis in the MSSM. We improve previously suggested benchmark
scenarios of maximal CP violation for analyzing direct searches of CP-violating
MSSM Higgs bosons at high-energy colliders, and stress the important
complementary r\^ole that a possible high-sensitivity measurement of the muon
electric dipole moment to the level of $10^{-24}$ $e$ cm can play in such
analyses.
