It is a classical result of Nirenberg and Pogorelov that a 2-sphere with positive Gauss curvature can be isometrically embedded in to Euclidean 3-space.
In this talk, I will present a proof due to Jiaxing Hong of a disc with positive Gauss curvature and whose boundary has positive geodesic curvature can be isometrically embedded into Euclidean half 3-space. If time permits, I will include a discussion of my own recent work on isometric
embeddings of 2-disc into hyperbolic space.