  We first extend the Peierls algebra of gauge invariant functions from the
space ${\cal S}$ of classical solutions to the space ${\cal H}$ of histories
used in path integration and some studies of decoherence. We then show that it
may be generalized in a number of ways to act on gauge dependent functions on
${\cal H}$. These generalizations (referred to as class I) depend on the choice
of an ``invariance breaking term," which must be chosen carefully so that the
gauge dependent algebra is a Lie algebra. Another class of invariance breaking
terms is also found that leads to an algebra of gauge dependent functions, but
only on the space ${\cal S}$ of solutions. By the proper choice of invariance
breaking term, we can construct a generalized Peierls algebra that agrees with
any gauge dependent algebra constructed through canonical or gauge fixing
methods, as well as Feynman and Landau ``gauge." Thus, generalized Peierls
algebras present a unified description of these techniques. We study the
properties of generalized Peierls algebras and their pull backs to spaces of
partial solutions and find that they may posses constraints similar to the
canonical case. Such constraints are always first class, and quantization may
proceed accordingly.
