Since astronomers first started detecting exoplanets, interest in possible exomoons soon followed. Moons could be habitable worlds in their own right but also influence the habitability of planets they orbit. Besides from habitability, discovering moons would provide rich insights into the formation and evolution of planetary systems, potentially revealing the banality/uniqueness of our own solar system’s architecture. For the last few years, I’ve led a project called the Hunt for Exomoons with Kepler project which has now surveyed over 300 planets. I’ll first discuss how we look for moons with attention to the statistical inferences. Then, we will dive into the results of our multi-year survey, with some important upper limits on the population of moons revealed in a recent paper. Finally, I will discuss Kepler-1625 - a Jupiter-sized planet orbiting a Sun-mass star. After initial hints in the Kepler data, recent HST follow-up by my team has revealed evidence for a Neptune-sized exomoon.