We present results on the mass, spin, and redshift distributions of the ten binary black hole mergers detected in the first and second observing runs completed by Advanced LIGO and Advanced Virgo. 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 such binaries is well approximated by models with no more than 1% of black holes more massive than $45 M{}_{\\odot}$, and a power law index of $\\alpha = {{1.6}^{+1.5}_{-1.7}}$ (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 flat or increasing with redshift with $88$% probability. Marginalizing over uncertainties in the BBH population, we find robust estimates of the BBH merger rate density of $R = {{53.2}^{+58.5}_{-28.8}}$ 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.