We identify 73 z {approx} 7 and 59 z {approx} 8 candidate galaxies in the reionization epoch, and use this large 26-29.4 AB mag sample of galaxies to derive very deep luminosity functions to < - 18 AB mag and the star formation rate (SFR) density at z {approx} 7 and z {approx} 8 (just 800 Myr and 650 Myr after recombination, respectively). The galaxy sample is derived using a sophisticated Lyman-break technique on the full two-year Wide Field Camera 3/infrared (WFC3/IR) and Advanced Camera for Surveys (ACS) data available over the HUDF09 ({approx}29.4 AB mag, 5{sigma}), two nearby HUDF09 fields ({approx}29 AB mag, 5{sigma}, 14 arcmin{sup 2}), and the wider area Early Release Science ({approx}27.5 AB mag, 5{sigma}, {approx}40 arcmin{sup 2}). The application of strict optical non-detection criteria ensures the contamination fraction is kept low (just {approx}7% in the HUDF). This very low value includes a full assessment of the contamination from lower redshift sources, photometric scatter, active galactic nuclei, spurious sources, low-mass stars, and transients (e.g., supernovae). From careful modeling of the selection volumes for each of our search fields, we derive luminosity functions for galaxies at z {approx} 7 and z {approx} 8 to < - 18 AB mag. The faint-end slopes {alpha} at z {approx} 7 and z {approx} 8 are uncertain but very steep at {alpha} = -2.01 {+-} 0.21 and {alpha} = -1.91 {+-} 0.32, respectively. Such steep slopes contrast to the local {alpha} {approx}> -1.4 and may even be steeper than that at z {approx} 4 where {alpha} = -1.73 {+-} 0.05. With such steep slopes ({alpha} {approx}< -1.7) lower luminosity galaxies dominate the galaxy luminosity density during the epoch of reionization. The SFR densities derived from these new z {approx} 7 and z {approx} 8 luminosity functions are consistent with the trends found at later times (lower redshifts). We find reasonable consistency with the SFR densities implied from reported stellar mass densities being only {approx}40% higher at z < 7. This suggests that (1) the stellar mass densities inferred from the Spitzer Infrared Array Camera (IRAC) photometry are reasonably accurate and (2) that the initial mass function at very high redshift may not be very different from that at later times.