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The discrete Laplace-Beltrami operator plays a prominent role in many Digital Geometry Processing applications ranging from denoising to parameterization, editing, and physical simulation. The standard discretization uses the cotangents of the angles in the immersed mesh which leads to a variety of numerical problems. We advocate use of the intrinsic Laplace-Beltrami operator. It satis- fies a local maximum principle, guaranteeing, e.g., that no flipped triangles can occur in parameterizations. It also leads to better conditioned linear systems. The intrinsic Laplace-Beltrami operator is based on an intrinsic Delaunay triangulation of the surface. We give an incremental algorithm to construct such triangulations together with an overlay structure which captures the relationship between the extrinsic and intrinsic triangulations. Using a variety of example meshes we demonstrate the numerical benefits of the intrinsic Laplace-Beltrami operator.