We present TeSMo, a framework for text-controlled scene-aware motion generation based on denoising diffusion models. Previous text-to-motion methods focus on characters in isolation without considering scenes due to the limited availability of datasets that include motion, text descriptions, and interactive scenes. Our approach begins with pre-training a scene-agnostic text-to-motion diffusion model, emphasizing goal-reaching constraints on large-scale motion-capture datasets. We then enhance this model with a scene-aware component, fine-tuned using data augmented with detailed scene information, including ground plane and object shapes. To facilitate this process, we embed annotated navigation and interaction motions within scenes. The proposed method produces realistic and diverse human-object interactions, such as navigation and sitting, in different scenes with various object shapes, orientations, initial body positions, and poses. Extensive experiments demonstrate that our approach surpasses prior techniques in terms of the plausibility of human-scene interaction, as well as the realism and variety of the generated motions. Code will be released upon publication of this work.