The proteolytic cleavages to the Notch protein that occur as a result of ligand binding. Ligand binding at the cell surface exposes an otherwise inaccessible cleavage site in the extracellular portion of Notch, which when cleaved releases a membrane-tethered form of the Notch intracellular domain. Subsequent cleavage within the transmembrane domain then leads to the release of the soluble Notch intracellular domain (NICD).