In this paper we introduce and study a new model for three–dimensional turbulence, the Leray–
α
model. This model is inspired by the Lagrangian averaged Navier–Stokes–
α
model of turbulence (also known Navier–Stokes–
α
model or the viscous Camassa–Holm equations). As in the case of the Lagrangian averaged Navier–Stokes–
α
model, the Leray–
α
model compares successfully with empirical data from turbulent channel and pipe flows, for a wide range of Reynolds numbers. We establish here an upper bound for the dimension of the global attractor (the number of degrees of freedom) of the Leray–
α
model of the order of (
L
/
l
d
)
12/7
, where
L
is the size of the domain and
l
d
is the dissipation length–scale. This upper bound is much smaller than what one would expect for three–dimensional models, i.e. (
L
/
l
d
)
3
. This remarkable result suggests that the Leray–
α
model has a great potential to become a good sub–grid–scale large–eddy simulation model of turbulence. We support this observation by studying, analytically and computationally, the energy spectrum and show that in addition to the usual
k
−5/3
Kolmogorov power law the inertial range has a steeper power–law spectrum for wavenumbers larger than 1/
α
. Finally, we propose a Prandtl–like boundary–layer model, induced by the Leray–
α
model, and show a very good agreement of this model with empirical data for turbulent boundary layers.