This paper studies a class of translation invariant states on quantum spin chains, characterized by the property that correlations across any bond can be modeled on a finite-dimensional vector space. These states are generalized valence bond (VBS) states and are dense in the set of all translation invariant states. The paper develops a complete theory of the ergodic decomposition of such states, including the decomposition into periodic "Néel ordered" states. The ergodic components have exponential decay of correlations. All these states can be obtained as "local functions" of states of a special kind, called "purely generated states," which are shown to be ground states for suitably chosen finite range VBS interactions. The paper shows that all these generalized VBS models have a spectral gap. The theory does not require symmetry of the state with respect to a local gauge group. The paper illustrates the results with a one-parameter family of examples, which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki. The paper discusses the difficulty of determining ground state properties of quantum spin systems on a lattice, which is much more complex than in classical statistical mechanics. In quantum mechanics, the restriction of a pure state is usually not pure, and a translation invariant pure state may have a rich structure of long range correlations. The paper aims to present and study a class of states and related Hamiltonians for which such questions can be answered explicitly. The paper also discusses the extension problem for quantum states, which is trivial for product states. The paper discusses the uniqueness of the finite volume ground state for a large class of models, including the standard nearest neighbor isotropic Heisenberg antiferromagnets on any finite bipartite lattice. The paper also discusses the occurrence of Néel order in translation and rotation invariant models, and the importance of considering systems in the thermodynamic limit to get relevant examples of symmetry breaking. The paper also discusses the conjecture by Haldane that the behavior of the ground states of one-dimensional antiferromagnetic nearest neighbor interactions depends qualitatively on whether the value of the spin s is integer or half-integer. The paper also discusses the exact solvability of several Hamiltonians by the Bethe Ansatz or Yang-Baxter type methods, and the determination of the ground state energy and the absence or existence of a gap. The paper also discusses the difficulty of determining correlation functions for these models. The paper also discusses the relative simplicity of obtaining correlation functions for another class of models, for which the ground states can be constructed exactly. These models are called VBS models, because of the Valence Bond structure of their ground states. The paper also discusses the Majumdar-Ghosh model, which has the same structure as VBS models, although the ground states are especially simple there. The states investigated in this paper are generalizations ofThis paper studies a class of translation invariant states on quantum spin chains, characterized by the property that correlations across any bond can be modeled on a finite-dimensional vector space. These states are generalized valence bond (VBS) states and are dense in the set of all translation invariant states. The paper develops a complete theory of the ergodic decomposition of such states, including the decomposition into periodic "Néel ordered" states. The ergodic components have exponential decay of correlations. All these states can be obtained as "local functions" of states of a special kind, called "purely generated states," which are shown to be ground states for suitably chosen finite range VBS interactions. The paper shows that all these generalized VBS models have a spectral gap. The theory does not require symmetry of the state with respect to a local gauge group. The paper illustrates the results with a one-parameter family of examples, which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki. The paper discusses the difficulty of determining ground state properties of quantum spin systems on a lattice, which is much more complex than in classical statistical mechanics. In quantum mechanics, the restriction of a pure state is usually not pure, and a translation invariant pure state may have a rich structure of long range correlations. The paper aims to present and study a class of states and related Hamiltonians for which such questions can be answered explicitly. The paper also discusses the extension problem for quantum states, which is trivial for product states. The paper discusses the uniqueness of the finite volume ground state for a large class of models, including the standard nearest neighbor isotropic Heisenberg antiferromagnets on any finite bipartite lattice. The paper also discusses the occurrence of Néel order in translation and rotation invariant models, and the importance of considering systems in the thermodynamic limit to get relevant examples of symmetry breaking. The paper also discusses the conjecture by Haldane that the behavior of the ground states of one-dimensional antiferromagnetic nearest neighbor interactions depends qualitatively on whether the value of the spin s is integer or half-integer. The paper also discusses the exact solvability of several Hamiltonians by the Bethe Ansatz or Yang-Baxter type methods, and the determination of the ground state energy and the absence or existence of a gap. The paper also discusses the difficulty of determining correlation functions for these models. The paper also discusses the relative simplicity of obtaining correlation functions for another class of models, for which the ground states can be constructed exactly. These models are called VBS models, because of the Valence Bond structure of their ground states. The paper also discusses the Majumdar-Ghosh model, which has the same structure as VBS models, although the ground states are especially simple there. The states investigated in this paper are generalizations of