Solid evidence of magnetic reconnection is rarely reported within sunspots, the darkest regions with the strongest magnetic fields and lowest temperatures in the solar atmosphere. Using the world's largest solar telescope, the 1.6 m Goode Solar Telescope, we detect prevalent reconnection through frequently occurring fine-scale jets in the Ha line wings at light bridges, the bright lanes that may divide the dark sunspot core into multiple parts. Many jets have an inverted Y-shape, shown by models to be typical of reconnection in a unipolar field environment. Simultaneous spectral imaging data from the Interface Region Imaging Spectrograph show that the reconnection drives bidirectional flows up to 200(.) km(.) s(-1), and that the weakly ionized plasma is heated by at least an order of magnitude up to similar to 80,000 K. Such highly dynamic reconnection jets and efficient heating should be properly accounted for in future modeling efforts of sunspots. Our observations also reveal that the surge-like activity previously reported above light bridges in some chromospheric passbands such as the Ha core has two components: the ever-present short surges likely to be related to the upward leakage of magnetoacoustic waves from the photosphere, and the occasionally occurring long and fast surges that are obviously caused by the intermittent reconnection jets.