We introduce GaussianAvatar, a novel approach for real-time, memory-efficient, high-quality animatable human modeling. GaussianAvatar takes as input a single monocular video to create a digital avatar capable of re-articulation in new poses and real-time rendering from novel viewpoints, while seamlessly integrating with rasterization-based graphics pipelines. Central to our method is the Gaussians-on-Mesh representation, a hybrid 3D model combining rendering quality and speed of Gaussian splatting with geometry modeling and compatibility of deformable meshes. We assess GaussianAvatar on ZJU-MoCap data and various YouTube videos. GaussianAvatar matches or surpasses current monocular human modeling algorithms in rendering quality and significantly outperforms them in computational efficiency (43 FPS) while being memory-efficient (3.63 MB per subject).