To elucidate the mechanisms of phenotypic variation and non-genetic inheritance in populations of genetically identical cells, we develop tools to study lineages of single cells and their progeny. Cells are trapped either in chambers as wide as a single cell and long enough to contain many generations of cells, or in 15 micron droplets of a water-in-oil emulsion generated in microfluidic devices. For lineage tracking, we hydrodynamically trap single cells in long, thin chambers, so they are constrained to grow in a line, making it easy to simultaneously follow many lineages deriving from single progenitor cells. Yeast cells grow and divide in the progeny chambers with doubling times similar to growth in liquid media up to about 5-6 generations. The cells remain immobilized in the chambers, providing a unique way to fix and stain multiple generations of yeast cells for in situ assays such as small molecule probing and immunofluorescence. Since all progeny deriving from the single cell are contained in the chamber, these new tools facilitate studies of gene expression, stress response, and DNA repair as a function of cell genealogy and replicative age. We also trap cells in 15 micron droplets of a water-in-oil emulsion generated in PDMS microfluidic devices, so cells remain isolated single cells over the experimental timecourse. We have developed a device to immobilize drops, allowing us to track single cell growth and to monitor levels of secreted enzymes that rapidly attain high concentrations due to the small drop volume.