We have measured the resistance vs. temperature of more than 20 superconducting nanowires with nominal widths ranging from 10 to 22 nm and lengths from 100 nm to 1050 nm. With decreasing cross-sectional areas, the wires display increasingly broad resistive transitions. The data are in very good agreement with a model that includes both thermally activated phase slips close to Tc and quantum phase slips (QPS) at low temperatures, but disagree with an earlier model based on a critical value of R_n/Rq. Our measurements provide strong evidence for QPS in thin superconducting wires.