Bayesian methods provide a principled approach to some of the hardest problems in computer vision—low signal-to-noise ratios, ill-posed problems, and problems with missing data. This dissertation applies Bayesian modeling to infer multidimensional continuous manifolds (e.g., curves, surfaces) from image data using Gaussian process priors. Gaussian processes are ideal priors in this setting, providing a stochastic model over continuous functions while permitting efficient inference. We begin by introducing a formal mathematical representation of branch curvilinear structures called a curve tree and we define a novel family of Gaussian processes over curve trees called branching Gaussian processes. We define two types of branching Gaussian properties and show how to extend them to branching surfaces and hypersurfaces. We then apply Gaussian processes in three computer vision applications. First, we perform 3D reconstruction of moving plants from 2D images. Using a branching Gaussian process prior, we recover high quality 3D trees while being robust to plant motion and camera calibration error. Second, we perform multi-part segmentation of plant leaves from highly occluded silhouettes using a novel Gaussian process model for stochastic shape. Our method obtains good segmentations despite highly ambiguous shape evidence and minimal training data. Finally, we estimate 2D trees from microscope images of neurons with highly ambiguous branching structure. We first fit a tree to a blurred version of the image where structure is less ambiguous. Then we iteratively deform and expand the tree to fit finer images, using a branching Gaussian process regularizing prior for deformation. Our method infers natural tree topologies despite ambiguous branching and image data containing loops. Our work shows that Gaussian processes can be a powerful building block for modeling complex structure, and they perform well in computer vision problems having significant noise and ambiguity.