In this paper we examine the cosmological consequences of fourth order Galileon gravity .
We carry out detailed investigations of the underlying dynamics and demonstrate the stability of one de Sitter phase .
The stable de Sitter phase contains a Galileon field \pi which is an increasing function of time ( \dot { \pi } > 0 ) .
Using the required suppression of the fifth force , supernovae , BAO , and CMB data , we constrain parameters of the model .
We find that the \pi matter coupling parameter \beta is constrained to small numerical values such that \beta < 0.02 .
We also show that the parameters of the third and fourth order in the action ( c _ { 3 } ,c _ { 4 } ) are not independent and with reasonable assumptions , we obtain constraints on them .
We investigate the growth history of the model and find that the sub-horizon approximation is not allowed for this model .
We demonstrate strong scale dependence of linear perturbations in the fourth order Galileon gravity .