Here I report on the hightlights of the ERC research performed in the duration of the ERC starting grant. In the papers "Weight one Jacobi forms and umbral moonshine" and in particular in "Optimal mock Jacobi theta functions", we have clarified the nature of the modular-like objects appearing in umbral moonshine. In "K3 string theory, lattices and moonshine", we made sharp conjectures on the relation between umbral moonshine and symmetries of string theory on K3, and provided non-trivial evidence for them. In "K3 Elliptic Genus and an Umbral Moonshine Module", we exploited the connection to string theory to construct the moonshine module for an important instance of umbral moonshine. In "Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules", we constructed the moonshine modules for another four instances of umbral moonshine by employing the expressions for the umbral moonshine functions in terms of meromorphic Jacobi forms. In "Vertex operator superalgebra/sigma model correspondences: The four-torus case", we extend and shed further light on the relation between K3 sigma models, umbral moonshine, and a particular vertex operator super-algebra by establishing an analogous relation in the case of T^4 sigma models. I believe that these papers and the talks our team has given on the topics have significantly furthered the progress of the research on this topic.