In this talk I will introduce the realization problem in persistence, which asks what isomorphism classes of diagrams of vector spaces can be realized by diagrams of topological spaces and continuous maps. In particular, one could ask what barcodes can be obtained by filtering a simplicial complex or by studying the level-set persistence of a map. I will review existing results for point clouds, and present some results of my own, obtained in collaboration with Ulrich Bauer and Hans Reiss. These results include studying the level-set barcodes realized by a stratified space and real-valued map, the space of barcodes realized by filtering a manifold by Gauss curvature, and the space of barcodes realized by Morse functions. To address which barcodes can be realized as the level-set barcodes of a Morse function $f$, I will present two constructions. One construction uses handlebody theory. The other construction is more novel and uses a cosheaf of spaces over the Reeb graph of $f$, which incidentally makes headway into a problem posed by Arnold. Additionally, this construction offers a vision for extending Mapper degree by degree (in analogy with Postnikov towers), offering a potentially powerful new tool in topological data analysis.