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 1 microm. 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 T(c) and quantum phase slips (QPS) at low temperatures, but disagree with an earlier model based on a critical value of R(N)/R(q). Our measurements provide strong evidence for QPS in thin superconducting wires.