The LIGO detection of GW150914 provides an unprecedented opportunity to study the two-bodymotion of a compact-object binary in the large-velocity, highly nonlinear regime, and to witness the finalmerger of the binary and the excitation of uniquely relativistic modes of the gravitational field.We carry outseveral investigations to determine whether GW150914 is consistent with a binary black-hole merger ingeneral relativity. We find that the final remnant’s mass and spin, as determined from the low-frequency(inspiral) and high-frequency (postinspiral) phases of the signal, are mutually consistent with the binaryblack-hole solution in general relativity. Furthermore, the data following the peak of GW150914 areconsistent with the least-damped quasinormal mode inferred from the mass and spin of the remnant blackhole. By using waveform models that allow for parametrized general-relativity violations during theinspiral and merger phases, we perform quantitative tests on the gravitational-wave phase in the dynamicalregime and we determine the first empirical bounds on several high-order post-Newtonian coefficients.Weconstrain the graviton Compton wavelength, assuming that gravitons are dispersed in vacuum in the sameway as particles with mass, obtaining a 90%-confidence lower bound of 1013 km. In conclusion, within ourstatistical uncertainties, we find no evidence for violations of general relativity in the genuinely strong-fieldregime of gravity.We report the observation of a gravitational-wave signal produced by the coalescence of two stellar-massblack holes. The signal, GW151226, was observed by the twin detectors of the Laser InterferometerGravitational-Wave Observatory (LIGO) on December 26, 2015 at 03:38:53 UTC. The signal was initiallyidentified within 70 s by an online matched-filter search targeting binary coalescences. Subsequent off-lineanalyses recovered GW151226 with a network signal-to-noise ratio of 13 and a significance greater than5σ. The signal persisted in the LIGO frequency band for approximately 1 s, increasing in frequency andamplitude over about 55 cycles from 35 to 450 Hz, and reached a peak gravitational strain of3.4+0.7−0.9 × 10^(−22). The inferred source-frame initial black hole masses are 14.2+8.3−3.7M⊙ and 7.5+2.3−2.3M⊙,and the final black hole mass is 20.8+6.1−1.7M⊙.We find that at least one of the component black holes has spingreater than 0.2. This source is located at a luminosity distance of 440+180−190 Mpc corresponding to a redshiftof 0.09+0.03−0.04 . All uncertainties define a 90% credible interval. This second gravitational-wave observationprovides improved constraints on stellar populations and on deviations from general relativity.