  This is a review devoted to the complementarity-contextuality interplay with
connection to the Bell inequalities. Starting discussion with complementarity,
we point out to contextuality as its seed. {\it Bohr-contextuality} is
dependence of observable's outcome on the experimental context, on
system-apparatus interaction. Probabilistically, complementarity means that the
{\it joint probability distribution} (JPD) does not exist. Instead of the JPD,
one has to operate with contextual probabilities. The Bell inequalities are
interpreted as the statistical tests of contextuality and, hence,
incompatibility. For context dependent probabilities, these inequalities may be
violated. We stress that contextuality tested by the Bell inequalities is so
called {\it joint measurement contextualit}y (JMC), the special case of Bohr's
contextuality. Then, we examine the role of signaling (marginal inconsistency).
In QM, signaling can be considered as an experimental artifact. However, often
experimental data has signaling patterns. We discuss possible sources of
signaling; for example, dependence of the state preparation on measurement
settings. In principle, one can extract the measure of "pure contextualize"
from data shadowed by signaling. This theory known as {\it Contextuality by
Default} (CbD). It leads to inequalities with the additional term quantifying
signaling, Bell-Dzhafarov-Kujala inequalities.
