We present DreamHuman, a method to generate realistic animatable 3D human avatar models solely from textual descriptions. Recent text-to-3D methods have made considerable strides in generation, but are still lacking in important aspects. Control and often spatial resolution remain limited, existing methods produce fixed rather than animated 3D human models, and anthropometric consistency for complex structures like people remains a challenge. DreamHuman connects large text-to-image synthesis models, neural radiance fields, and statistical human body models in a novel modeling and optimization framework. This makes it possible to generate dynamic 3D human avatars with high-quality textures and learned, instance-specific, surface deformations. We demonstrate that our method is capable to generate a wide variety of animatable, realistic 3D human models from text. Our 3D models have diverse appearance, clothing, skin tones and body shapes, and significantly outperform both generic text-to-3D approaches and previous text-based 3D avatar generators in visual fidelity.