Big bang nucleosynthesis (BBN) describes the production of the lightest nuclides via a dynamic interplay among the four fundamental forces during the first seconds of cosmic time. A brief overview of the essentials of this physics is given, and new calculations presented of light-element abundances through Li6 and Li7, with updated nuclear reactions and uncertainties including those in the neutron lifetime. Fits are provided for these results as a function of baryon density and of the number of neutrino flavors Nν. Recent developments are reviewed in BBN, particularly new, precision Planck cosmic microwave background (CMB) measurements that now probe the baryon density, helium content, and the effective number of degrees of freedom Neff. These measurements allow for a tight test of BBN and cosmology using CMB data alone. Our likelihood analysis convolves the 2015 Planck data chains with our BBN output and observational data. Adding astronomical measurements of light elements strengthens the power of BBN. A new determination of the primordial helium abundance is included in our likelihood analysis. New D/H observations are now more precise than the corresponding theoretical predictions and are consistent with the standard model and the Planck baryon density. Moreover, D/H now provides a tight measurement of Nν when combined with the CMB baryon density and provides a 2σ upper limit Nν<3.2. The new precision of the CMB and D/H observations together leaves D/H predictions as the largest source of uncertainties. Future improvement in BBN calculations will therefore rely on improved nuclear cross-section data. In contrast with D/H and He4, Li7 predictions continue to disagree with observations, perhaps pointing to new physics. This paper concludes with a look at future directions including key nuclear reactions, astronomical observations, and theoretical issues.