The rapidly growing statistical precision of galaxy surveys has lead to a need for ever-more precise predictions of the observables used to constrain cosmological and galaxy formation models. The primary avenue through which such predictions will be obtained is suites of numerical simulations. These simulations must span the relevant model parameter spaces, be large enough to obtain the precision demanded by upcoming data, and be thoroughly validated in order to ensure accuracy. In this paper

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... present one such suite of simulations, forming the basis for the AEMULUS Project, a collaboration devoted to precision emulation of galaxy survey observables. We have run a set of 75 (1.05 h^-1 Gpc)^3 simulations with mass resolution and force softening of 3.51\times 10^10 (Omega_m / 0.3) ~ h^-1 M_sun and 20 ~ h^-1 kpc respectively in 47 different wCDM cosmologies spanning the range of parameter space allowed by the combination of recent Cosmic Microwave Background, Baryon Acoustic Oscillation and Type Ia Supernovae results. We present convergence tests of several observables including spherical overdensity halo mass functions, galaxy projected correlation functions, galaxy clustering in redshift space, and matter and halo correlation functions and power spectra. We show that these statistics are converged to 1% (2%) for halos with more than 500 (200) particles respectively and scales of r>200 ~ h^-1 kpc in real space or k ~ 3 h Mpc^-1 in harmonic space for z\le 1. We find that the dominant source of uncertainty comes from varying the particle loading of the simulations. This leads to large systematic errors for statistics using halos with fewer than 200 particles and scales smaller than k ~ 4 h^-1 Mpc. We provide the halo catalogs and snapshots detailed in this work to the community at https://AemulusProject.github.io.