Gibbs measures, as used in Statistical Mechanics, have a definition that is remarkably similar to the definition ofg-measures, used in dynamical systems. For both types of measures the continuity of conditional probabilities play a central role.In this thesis we give necessary and sufficient conditions for when a g-measure is a Gibbs measure. We relate this result to well known uniqueness conditions and briefly consider the related question: when is a g-measure reversible.Subsequently we consider an application in information theory by considering whether one-sided models can be used for two-sided modeling.Finally, we apply a technique called measure disintegration to give very general conditions for when the conditional probabilities of factors of Markov processesare g-measures and factors of Gibbs measure are Gibbs measures.