  We give an elementary introduction to Classical Invariant Theory and its
modern extension "Transcending Classical Invariant Theory", commonly known as
the theory of local theta correspondence. We explain the two fundamental
assertions of the theory: the Howe duality conjecture and the Kudla-Rallis
conservation relation conjecture. We give a status report on the problem of
explicitly describing local theta correspondence in terms of Langlands-Vogan
parameters. We conclude with a discussion on a certain problem of automatic
continuity, which manifests unity of the theory in algebraic and smooth
settings.
