  We introduce a class of finite dimensional nonlinear superalgebras $L =
L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq
sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of
degree $>1$) in the $gl(n)$ generators. Specifically, we investigate `type I'
super-$gl(n)$ algebras, having odd generators transforming in a single
irreducible representation of $gl(n)$ together with its contragredient.
Admissible structure constants are discussed in terms of available $gl(n)$
couplings, and various special cases and candidate superalgebras are identified
and exemplified via concrete oscillator constructions. For the case of the
$n$-dimensional defining representation, with odd generators $Q_{a},
\bar{Q}{}^{b}$, and even generators ${E^{a}}_{b}$, $a,b = 1,...,n$, a three
parameter family of quadratic super-$gl(n)$ algebras (deformations of
$sl(n/1)$) is defined. In general, additional covariant Serre-type conditions
are imposed, in order that the Jacobi identities be fulfilled. For these
quadratic super-$gl(n)$ algebras, the construction of Kac modules, and
conditions for atypicality, are briefly considered. Applications in quantum
field theory, including Hamiltonian lattice QCD and space-time supersymmetry,
are discussed.
