  We introduce the Virgo Consortium's FLAMINGO suite of hydrodynamical
simulations for cosmology and galaxy cluster physics. To ensure the simulations
are sufficiently realistic for studies of large-scale structure, the subgrid
prescriptions for stellar and AGN feedback are calibrated to the observed
low-redshift galaxy stellar mass function and cluster gas fractions. The
calibration is performed using machine learning, separately for three
resolutions. This approach enables specification of the model by the
observables to which they are calibrated. The calibration accounts for a number
of potential observational biases and for random errors in the observed stellar
masses. The two most demanding simulations have box sizes of 1.0 and 2.8 Gpc
and baryonic particle masses of $1\times10^8$ and $1\times10^9 \text{M}_\odot$,
respectively. For the latter resolution the suite includes 12 model variations
in a 1 Gpc box. There are 8 variations at fixed cosmology, including shifts in
the stellar mass function and/or the cluster gas fractions to which we
calibrate, and two alternative implementations of AGN feedback (thermal or
jets). The remaining 4 variations use the unmodified calibration data but
different cosmologies, including different neutrino masses. The 2.8 Gpc
simulation follows $3\times10^{11}$ particles, making it the largest ever
hydrodynamical simulation run to $z=0$. Lightcone output is produced on-the-fly
for up to 8 different observers. We investigate numerical convergence, show
that the simulations reproduce the calibration data, and compare with a number
of galaxy, cluster, and large-scale structure observations, finding very good
agreement with the data for converged predictions. Finally, by comparing
hydrodynamical and `dark-matter-only' simulations, we confirm that baryonic
effects can suppress the halo mass function and the matter power spectrum by up
to $\approx20$ per cent.
