  We introduce the TangoSIDM project, a suite of cosmological simulations of
structure formation in a $\Lambda$-Self-Interacting Dark Matter (SIDM)
universe. TangoSIDM explores the impact of large dark matter (DM) scattering
cross sections over dwarf galaxy scales. Motivated by DM interactions that
follow a Yukawa potential, the cross section per unit mass, $\sigma/m_{\chi}$,
assumes a velocity dependent form that avoids violations of current constraints
on large scales. We demonstrate that our implementation accurately models not
only core formation in haloes, but also gravothermal core collapse. For central
haloes in cosmological volumes, frequent DM particle collisions isotropise the
particles orbit, making them largely spherical. We show that the
velocity-dependent $\sigma/m_{\chi}$ models produce a large diversity in the
circular velocities of satellites haloes, with the spread in velocities
increasing as the cross sections reach 20, 60 and 100 cm$^2$/g in
$10^9~\rm{M}_{\odot}$ haloes. The large variation in the haloes internal
structure is driven by DM particles interactions, causing in some haloes the
formation of extended cores, whereas in others gravothermal core collapse. We
conclude that the SIDM models from the Tango project offer a promising
explanation for the diversity in the density and velocity profiles of observed
dwarf galaxies.
