We present results on the mass, spin, and redshift distributions of the ten binary black hole mergers detected in Advanced LIGO's and Advanced Virgo's first and second observing runs. We constrain properties of the binary black hole (BBH) mass spectrum using models with a range of parameterizations of the BBH mass and spin distributions. We find that the mass distribution of the more massive black hole in each binary is well approximated by models with almost no black holes larger than 45M⊙, and a power law index of α=1.6+1.5−1.7 (90% credibility). We also show that BBHs are unlikely to be composed of black holes with large spins aligned to the orbital angular momentum. Modelling the evolution of the BBH merger rate with redshift, we show that it is increasing with redshift with credibility 0.88. Marginalizing over uncertainties in the BBH population, we find robust estimates of the BBH merger rate density of R=52.9+55.6−27.0 Gpc−3 yr−1 (90% credibility). As the BBH catalog grows in future observing runs, we expect that uncertainties in the population model parameters will shrink, potentially providing insights into the formation of black holes via supernovae, binary interactions of massive stars, stellar cluster dynamics, and the formation history of black holes across cosmic time.