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The methods developed in the thesis are largely motivated by one particular application. The function of transplanted lungs often deteriorates through the years after the transplant, usually attributed to a condition known as bronchiolitis obliterans syndrome. The presence of the underlying disease can only be strictly confirmed by an invasive biopsy procedure. Therefore, in practice, lung function is determined by repeated measurements of forced expiratory volume in one second. A set of data from Papworth Hospital's transplant programme is used to illustrate many of the models. Staged progression can be described using multi-state models on Markov processes. These are initially described in contexts where each patient's true status is known at a series of observation times. Extensions to these models are presented for situations where the underlying disease stage is not observed. These hidden Markov models can either be based on repeated observations of a disease stage with error, or repeated observations of the continuous biomarker. Methods illustrated for fitting these models to data include maximum likelihood and Bayesian estimation. The main contrasting approach directly models the evolution of the continuous process through time using longitudinal markers. A general model is presented, which can include random effects growth curves, continuous stochastic evolution, and semi-parametric estimation of marker trajectories. Various special cases of this model are illustrated on the lung function marker data. Finally, a hierarchical Bayesian model is developed which can encompass different underlying patterns of disease evolution among a set of patients. It is based on finite population mixtures. In the illustrative application, disease onset can either occur suddenly, inducing change-point in the biomarker trajectory, or smoothly, described by a linear model for the marker. Markov chain Monte Carlow techniques allow the simultaneous Bayesian estimation of all unknown quantities in the hierarchy.