In this work we address the task of segmenting an object into its parts, or semantic part segmentation. We start by adapting a state-of-the-art semantic segmentation system to this task, and show that a combination of a fully-convolutional Deep CNN system coupled with Dense CRF labelling provides excellent results for a broad range of object categories. Still, this approach remains agnostic to high-level constraints between object parts. We introduce such prior information by means of the Restricted Boltzmann Machine, adapted to our task and train our model in an discriminative fashion, as a hidden CRF, demonstrating that prior information can yield additional improvements. We also investigate the performance of our approach ``in the wild'', without information concerning the objects' bounding boxes, using an object detector to guide a multi-scale segmentation scheme. We evaluate the performance of our approach on the Penn-Fudan and LFW datasets for the tasks of pedestrian parsing and face labelling respectively. We show superior performance with respect to competitive methods that have been extensively engineered on these benchmarks, as well as realistic qualitative results on part segmentation, even for occluded or deformable objects. We also provide quantitative and extensive qualitative results on three classes from the PASCAL Parts dataset. Finally, we show that our multi-scale segmentation scheme can boost accuracy, recovering segmentations for finer parts.