We develop spectral theory for the generator of the q-Boson particle system. Our central result is a Plancherel type isomorphism theorem for this system; it implies completeness of the Bethe ansatz in infinite volume and enables us to solve forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. Degenerations also imply similar (partially previously known) results for the delta Bose gas and its discretizations. Familiarity with the content of previous week's lectures would be helpful for seeing the `big picture', but not necessary as this talk will be essentially independent.