  We have recently proposed a novel self tuning mechanism to alleviate the
famous cosmological constant problem, based on the general scalar tensor theory
proposed by Horndeski. The self-tuning model ends up consisting of four
geometric terms in the action, with each term containing a free potential
function of the scalar field; the four together being labeled as the Fab-Four.
In this paper we begin the important task of deriving the cosmology associated
with the Fab-Four Lagrangian. Performing a phase plane analysis of the system
we are able to obtain a number of fixed points for the system, with some
remarkable new solutions emerging from the trade-off between the various
potentials. As well as obtaining inflationary solutions we also find
conventional radiation/matter-like solutions, but in regimes where the energy
density is dominated by a cosmological constant, and where we do not have any
explicit forms of radiation or matter. Stability conditions for matter
solutions are obtained and we show how it is possible for there to exist an
extended period of `matter domination' opening up the possibility that we can
generate cosmological structures, and recover a consistent cosmology even in
the presence of a large cosmological constant.
