We investigate the structure of complete locally irreducible conullity two Riemannian manifolds. If ( \Gamma ) is the nullity space of the curvature tensor of a complete manifold ( M^n ) and both ( n ) and ( \dim \Gamma ) are constant then ( \Gamma ) is completely integrable with flat leaves. Utilizing tools such as the splitting tensor we arrive at a natural classification of two potential cases: the so-called nilpotent conullity two manifolds whose splitting tensors are nilpotent, and the hyperbolic conullity two manifolds whose splitting tensors have complex conjugate non-real eigenvalues. In the nilpotent case we arrive at a Lipschitz foliation by totally geodesic flat hyperplanes and determine the metric on an open dense subset in terms of ( n-1 ) functions. We also find examples where the foliation is only smooth on specific subsets, and we determine that the fundamental group must either be trivial or infinite cyclic. In the hyperbolic case we make some observations that could lead to the development of a classification.