Geometric models play a vital role in several fields, from the entertainment industry to scientific applications. To reduce the high cost of model creation, reusing existing models is the solution of choice. Model reuse is supported by content-based shape retrieval (CBR) techniques that help finding the desired models in massive repositories, many publicly available on the Internet. Key to efficient and effective CBR techniques are shape descriptors that accurately capture the characteristics of a shape and can discriminate between different shapes. We present a descriptor based on the distribution of two global features measured in a 3D shape, depth complexity and thickness, which respectively capture aspects of the geometry and topology of 3D shapes. The final descriptor, called DCTH (depth complexity and thickness histogram), is a 2D histogram that is invariant to the translation, rotation and scale of geometric shapes. We efficiently implement the DCTH on the GPU, allowing its use in real-time queries of large model databases. We validate the DCTH with the Princeton and Toyohashi Shape Benchmarks, containing 1815 and 10000 models respectively. Results show that DCTH can discriminate meaningful classes of these benchmarks and is fast to compute and robust against shape transformations and different levels of subdivision and smoothness.