This paper addresses the problem of detecting and segmenting partially occluded objects of a known category. We first define a part labelling which densely covers the object. Our Layout Consistent Random Field (LayoutCRF) model then imposes asymmetric local spatial constraints on these labels to ensure the consistent layout of parts whilst allowing for object deformation. Arbitrary occlusions of the object are handled by avoiding the assumption that the whole object is visible. The resulting system is both efficient to train and to apply to novel images, due to a novel annealed layout-consistent expansion move algorithm paired with a randomised decision tree classifier. We apply our technique to images of cars and faces and demonstrate state-of-the-art detection and segmentation performance even in the presence of partial occlusion.