The Apache Point Observatory Galactic Evolution Experiment (APOGEE) is obtaining high resolution ( 22,500), high signal-to-noise (> 100) spectra in the 1510-1690 nm spectral region for 100,000 cool, predominantly post-main sequence stars. To ascertain the stellar atmospheric parameters and measure chemical abundances for the numerous chemical elements with line transitions in this wavelength region, the APOGEE Atmospheric Parameters and Chemical Abundances Pipeline (ASPCAP) relies on an optimization algorithm that identifies the best-fitting model for each of the observed APOGEE spectra. The fitting algorithm speeds up the model evaluation by interpolation in pre-computed grids of synthetic spectra that have been compressed using Principal Component Analysis. Here we describe the main model grids used in ASPCAP for the tenth data release of the Sloan Digital Sky Survey (SDSS DR10), how they were calculated. We also provide a description of ongoing and planned upgrades.