We perform a search for stellar streams around the Milky Way using the first 3 yr of multiband optical imaging data from the Dark Energy Survey (DES). We use DES data covering ∼5000 deg 2 to a depth of g>23.5 with a relative photometric calibration uncertainty of <1%. This data set yields unprecedented sensitivity to the stellar density field in the southern celestial hemisphere, enabling the detection of faint stellar streams to a heliocentric distance of ∼50 kpc. We search for stellar streams using a matched filter in color-magnitude space derived from a synthetic isochrone of an old, metal-poor stellar population. Our detection technique recovers four previously known thin stellar streams: Phoenix, ATLAS, Tucana III, and a possible extension of Molonglo. In addition, we report the discovery of 11 new stellar streams. In general, the new streams detected by DES are fainter, more distant, and lower surface brightness than streams detected by similar techniques in previous photometric surveys. As a by-product of our stellar stream search, we find evidence for extratidal stellar structure associated with four globular clusters: NGC 288, NGC 1261, NGC 1851, and NGC 1904. The ever-growing sample of stellar streams will provide insight into the formation of the Galactic stellar halo, the Milky Way gravitational potential, and the large-and small-scale distribution of dark matter around the Milky Way.