This thesis introduces Gaussian process dynamical models (GPDMs) for nonlinear time series analysis. A GPDM comprises a low-dimensional latent space with associated dynamics, and a map from the latent space to an observation space. We marginalize out the model parameters in closed-form, which leads to modeling both dynamics and observation mappings as Gaussian processes. This results in a nonparametric model for dynamical systems that accounts for uncertainty in the model. We train the model on human motion capture data in which each pose is 62-dimensional, and synthesize new motions by sampling from the posterior distribution. A comparison of forecasting results between different covariance functions and sampling methods is provided, and we demonstrate a simple application of GPDM on filling in missing data. Finally, to account for latent space uncertainty, we explore different priors settings on hyperparameters and show some preliminary GPDM learning results using a Monte Carlo expectation-maximization algorithm.