  Self-consistent ${\it N}$-body simulations of modified gravity models are a
key ingredient to obtain rigorous constraints on deviations from General
Relativity using large-scale structure observations. This paper provides the
first detailed comparison of the results of different ${\it N}$-body codes for
the $f(R)$, DGP, and Symmetron models, starting from the same initial
conditions. We find that the fractional deviation of the matter power spectrum
from $\Lambda$CDM agrees to better than $1\%$ up to $k \sim 5-10~h/{\rm Mpc}$
between the different codes. These codes are thus able to meet the stringent
accuracy requirements of upcoming observational surveys. All codes are also in
good agreement in their results for the velocity divergence power spectrum,
halo abundances and halo profiles. We also test the quasi-static limit, which
is employed in most modified gravity ${\it N}$-body codes, for the Symmetron
model for which the most significant non-static effects among the models
considered are expected. We conclude that this limit is a very good
approximation for all of the observables considered here.
