  We revisit the study of the multiplets of the conformal algebra in any
dimension. The theory of highest weight representations is reviewed in the
context of the Bernstein-Gelfand-Gelfand category of modules. The
Kazhdan-Lusztig polynomials code the relation between the Verma modules and the
irreducible modules in the category and are the key to the characters of the
conformal multiplets (whether finite dimensional, infinite dimensional, unitary
or non-unitary). We discuss the representation theory and review in full
generality which representations are unitarizable. The mathematical theory that
allows for both the general treatment of characters and the full analysis of
unitarity is made accessible. A good understanding of the mathematics of
conformal multiplets renders the treatment of all highest weight
representations in any dimension uniform, and provides an overarching
comprehension of case-by-case results. Unitary highest weight representations
and their characters are classified and computed in terms of data associated to
cosets of the Weyl group of the conformal algebra. An executive summary is
provided, as well as look-up tables up to and including rank four.
