For robots to perform intricate manipulation skills, like picking up a slippery banana slice with a fork, it is often useful to have a human demonstrate how to perform that skill for the robot. Humans can perform the desired motion multiple times in front of the robot, and the robot can record the demonstrated trajectories and build a model of the demonstrations. If the robot can learn a good model of the different ways to perform the desired motion, the human and the robot can then work together to pick a trajectory for the robot to perform to solve the task. This dissertation investigates the machine learning component of that example: "How can a robot learn a good model of demonstrated trajectories?" We present multiple advances in the ability of robots to model demonstrated trajectories using latent variable models. These approaches include better model regularization to take advantage of the small size of datasets of human demonstrations, better architectural choices to separate the timing and spatial variations of the demonstrated trajectories, and an investigation into how to disentangle the meaning of the variables in the latent variable model. Theoretical justifications for the contributions are presented alongside empirical evaluations performed on a physical robot arm.