  Here we examine O(n) systems with arbitrary two spin interactions (of
unspecified range) within a general framework. We shall focus on
translationally invariant interactions. In the this case, we determine the
ground states of the $O(n \ge 2)$ systems. We further illustrate how one may
establish Peierls bounds for many Ising systems with long range interactions.
We study the effect of thermal fluctuations on the ground states and derive the
corresponding fluctuation integrals. The study of the thermal fluctuation
spectra will lead us to discover a very interesting odd-even $n$
(coupling-decoupling) effect. We will prove a generalized Mermin-Wagner-Coleman
theorem for all two dimensional systems (of arbitrary range) with analytic
kernels in $k$ space. We will show that many three dimensional systems have
smectic like thermodynamics. We will examine the topology of the ground state
manifolds for both translationally invariant and spin glass systems. We
conclude with a discussion of O(n) spin dynamics in the general case.
