Gaussian SLAM is capable of reconstructing renderable 3D scenes from RGBD data streams. It is the first neural RGBD SLAM method capable of reconstructing real-world scenes with photorealistic fidelity. By leveraging 3D Gaussian as the primary unit for scene representation, we overcome the limitations of previous methods. We observe that traditional 3D Gaussians are difficult to utilize in monocular settings: they fail to encode accurate geometric information and are challenging to optimize sequentially with single-view supervision. By extending traditional 3D Gaussians to encode geometric information and designing a novel scene representation as well as a method for its growth and optimization, we propose an SLAM system that can reconstruct and render real-world datasets while maintaining speed and efficiency. Gaussian SLAM is able to reconstruct and render real-world scenes with photorealistic fidelity. We evaluate our method on common synthetic and real-world datasets, comparing it against other state-of-the-art SLAM methods. Finally, we demonstrate that the resulting 3D scene representation can be efficiently rendered in real-time using Gaussian splatting.