  We propose a new approach to a unified study of determinants, permanents,
immanants, (determinantal) bitableaux and symmetrized bitableaux in the
polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the
enveloping algebra $U(gl(n))$. This leads to new relevant classes of elements
in $U(gl(n))$: Capelli bitableaux, right Young-Capelli bitableaux and Capelli
immanants. The set of standard Capelli bitableaux and the set of standard right
Young-Capelli bitableaux are bases of $U(gl(n))$, whose action on the
Gordan-Capelli basis of $C[M_{n, n}]$ have remarkable properties. Capelli
immanants can be efficiently computed and provide a system of generators of
$U(gl(n))$. The Okounkov quantum immanants are proved to be simple linear
combinations of Capelli immanants. Several examples are provided throughout the
paper.
