Protease that catalyzes two essential functions in the SUMO pathway: processing of full-length SUMOs to their mature forms and deconjugation of SUMO from targeted proteins. Cleaves precursors of SUM1 and SUM2, and very inefficiently of SUM3. Seems to be the only ULP1 able to cleave SUM3 precursors. Cleaves SUMO peptides better than SUMO-conjugated proteins. The N-terminal regulatory domain is required for peptidase activity in vitro. Belongs to the peptidase C48 family.