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Fuzzy Dark Matter (FDM), consisting of ultralight bosons, is an intriguing alternative to Cold Dark Matter (CDM). Unlike in CDM, FDM halos consist of a central solitonic core, surrounded by an envelope of order unity density fluctuations. The envelope density fluctuations also interact with the soliton causing it to wobble and oscillate. Using high-resolution numerical simulations of an FDM halo, corresponding to a particular boson mass, I will demonstrate that the gravitational potential fluctuations associated with the soliton's wobble, its oscillations, and the envelope density fluctuations dynamically heat nuclear objects (e.g., central star clusters and supermassive black holes) and galaxies. As a result, nuclear objects, initially located at rest at the soliton center, migrate outwards over time until the outward motion is counteracted by dynamical friction and an equilibrium is reached. Similarly, a galaxy undergoes significant size expansion and central density reduction over a Hubble time. Generalizing these results for other halo and boson masses and comparing them with observations (such as galaxy size-age relation, measured offsets of supermassive black holes and nuclear star clusters from the centers of their host galaxies) will be able to constrain the boson mass. After discussing FDM, I will also briefly present my work on the peculiar galaxy NGC 1052-DF2 and show what we can learn about its mass distribution from the dynamical friction-induced orbital decay of its globular clusters.