NASA's Kepler mission identified over 4,000 extrasolar planets that transit (cross in front of) their host stars. This sample has revealed detailed features in the demographics of planet sizes and orbital spacings. However, knowledge of their orbital shapes-a key tracer of planetary formation and evolution-remains far more limited. We present measurements of eccentricities for 1,646 Kepler planets, 92% of which are smaller than Neptune. For all planet sizes, the eccentricity distribution peaks at