The present invention is directed to methods for the expansion of non-terminally differentiated cells (“precursor cells”) using agonists of Notch function, by inhibiting the differentiation of the cells without inhibiting proliferation (mitotic activity) such that an expanded population of non-terminally differentiated cells is obtained. The cells are preferably stem or progenitor cells. These expanded cells can be used in cell replacement therapy to provide desired cell populations and help in the regeneration of diseased and/or injured tissues. The expanded cell populations can also be made recombinant and used for gene therapy, or can be used to supply functions associated with a particular precursor cell or its progeny cell.