Tutte embedding is one of the most common building blocks in geometry processing algorithms due to its simplicity and provable guarantees. Although provably correct in infinite precision arithmetic, it fails in challenging cases when implemented using floating point arithmetic, largely due to the induced exponential area changes. We propose Progressive Embedding, with similar theoretical guarantees to Tutte embedding, but more resilient to the rounding error of floating point arithmetic. Inspired by progressive meshes, we collapse edges on an invalid embedding to a valid, simplified mesh, then insert points back while maintaining validity. We demonstrate the robustness of our method by computing embeddings for a large collection of disk topology meshes. By combining our robust embedding with a variant of the matchmaker algorithm, we propose a general algorithm for the problem of mapping multiply connected domains with arbitrary hard constraints to the plane, with applications in texture mapping and remeshing.