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Methods to rapidly assess cell growth would be useful for many applications, including drug susceptibility testing, but current technologies have limited sensitivity or throughput. Here we present an approach to precisely and rapidly measure growth rates of many individual cells simultaneously. We flow cells in suspension through a microfluidic channel with 10-12 resonant mass sensors distributed along its length, weighing each cell repeatedly over the 4-20 min it spends in the channel. Because multiple cells traverse the channel at the same time, we obtain growth rates for >60 cells/h with a resolution of 0.2 pg/h for mammalian cells and 0.02 pg/h for bacteria. We measure the growth of single lymphocytic cells, mouse and human T cells, primary human leukemia cells, yeast, Escherichia coli and Enterococcus faecalis. Our system reveals subpopulations of cells with divergent growth kinetics and enables assessment of cellular responses to antibiotics and antimicrobial peptides within minutes.