We present the results from an all-sky search for short-duration gravitational waves in the data of the firstrun of the Advanced LIGO detectors between September 2015 and January 2016. The search algorithmsuse minimal assumptions on the signal morphology, so they are sensitive to a wide range of sourcesemitting gravitational waves. The analyses target transient signals with duration ranging from millisecondsto seconds over the frequency band of 32 to 4096 Hz. The first observed gravitational-wave event,GW150914, has been detected with high confidence in this search; the other known gravitational-waveevent, GW151226, falls below the search’s sensitivity. Besides GW150914, all of the search results areconsistent with the expected rate of accidental noise coincidences. Finally, we estimate rate-density limitsfor a broad range of non-binary-black-hole transient gravitational-wave sources as a function of theirgravitational radiation emission energy and their characteristic frequency. These rate-density upper limitsare stricter than those previously published by an order of magnitude.