  We present the Cosmology and Astrophysics with MachinE Learning Simulations
--CAMELS-- project. CAMELS is a suite of 4,233 cosmological simulations of
$(25~h^{-1}{\rm Mpc})^3$ volume each: 2,184 state-of-the-art
(magneto-)hydrodynamic simulations run with the AREPO and GIZMO codes,
employing the same baryonic subgrid physics as the IllustrisTNG and SIMBA
simulations, and 2,049 N-body simulations. The goal of the CAMELS project is to
provide theory predictions for different observables as a function of cosmology
and astrophysics, and it is the largest suite of cosmological
(magneto-)hydrodynamic simulations designed to train machine learning
algorithms. CAMELS contains thousands of different cosmological and
astrophysical models by way of varying $\Omega_m$, $\sigma_8$, and four
parameters controlling stellar and AGN feedback, following the evolution of
more than 100 billion particles and fluid elements over a combined volume of
$(400~h^{-1}{\rm Mpc})^3$. We describe the simulations in detail and
characterize the large range of conditions represented in terms of the matter
power spectrum, cosmic star formation rate density, galaxy stellar mass
function, halo baryon fractions, and several galaxy scaling relations. We show
that the IllustrisTNG and SIMBA suites produce roughly similar distributions of
galaxy properties over the full parameter space but significantly different
halo baryon fractions and baryonic effects on the matter power spectrum. This
emphasizes the need for marginalizing over baryonic effects to extract the
maximum amount of information from cosmological surveys. We illustrate the
unique potential of CAMELS using several machine learning applications,
including non-linear interpolation, parameter estimation, symbolic regression,
data generation with Generative Adversarial Networks (GANs), dimensionality
reduction, and anomaly detection.
