  The exploration of the notion of observability exhibits transparently the
rich interplay between algebraic and geometric ideas in \emph{geometric
invariant theory}. The concept of \emph{observable subgroup} was introduced in
the early 1960s with the purpose of studying extensions of representations from
an affine algebraic subgroup to the whole group. The extent of its importance
in \emph{representation and invariant theory} in particular for Hilbert's
$14^{\text{th}}$ problem was noticed almost immediately. An important
strenghtening appeared in the mid 1970s when the concept of \emph{strong
observability} was introduced and it was shown that the notion of observability
can be understood as an intermediate step in the notion of reductivity (or
semisimplicity), when adequately generalized. More recently starting in 2010,
the concept of observable subgroup was expanded to include the concept of
\emph{observable action} of an affine algebraic group on an affine variety,
launching a series of new applications. In 2006 the related concept of
\emph{observable adjunction} was introduced, and its application to module
categories over tensor categories was noticed. In the current survey, we follow
(approximately) the historical development of the subject introducing along the
way, the definitions and some of the main results including some of the proofs.
For the unproven parts, precise references are mentioned.
