  We study certain Z_2-graded, finite-dimensional polynomial algebras of degree
2 which are a special class of deformations of Lie superalgebras, which we call
quadratic Lie superalgebras. Starting from the formal definition, we discuss
the generalised Jacobi relations in the context of the Koszul property, and
give a proof of the PBW basis theorem. We give several concrete examples of
quadratic Lie superalgebras for low dimensional cases, and discuss aspects of
their structure constants for the `type I' class. We derive the equivalent of
the Kac module construction for typical and atypical modules, and a related
direct construction of irreducible modules due to Gould. We investigate in
detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie
superalgebra sl(n/1). We formulate the general atypicality conditions at level
1, and present an analysis of zero-and one-step atypical modules for a certain
family of Kac modules.
