The impact of plasma shaping on tokamak scrape-off layer (SOL) turbulence is investigated. The drift-reduced Braginskii equations are written for arbitrary magnetic geometries, and an analytical equilibrium model is used to introduce the dependence of turbulence equations on tokamak inverse aspect ratio (epsilon), Shafranov's shift (Delta), elongation (kappa), and triangularity (delta). A linear study of plasma shaping effects on the growth rate of resistive ballooning modes (RBMs) and resistive drift waves (RDWs) reveals that RBMs are strongly stabilized by elongation and negative triangularity, while RDWs are only slightly stabilized in non-circular magnetic geometries. Assuming that the linear instabilities saturate due to nonlinear local flattening of the plasma gradient, the equilibrium gradient pressure length Lp = -p(e)/del p(e) in the SOL is numerically computed and its dependence on epsilon, Delta, kappa and delta is analyzed, showing that stabilization of RBMs results in shorter Lp. An analytical estimate of L-p in the infinit aspect ratio limit and neglecting the Shafranov's shift is also derived. Nonlinear SOL turbulence simulations with non-circular magnetic geometries are carried out using the global, three-dimensional, flux-driven fluid code GBS (Ricci et al 2012 Plasma Phys. Control. Fusion 54 124047) and the results are compared with the findings obtained from the linear analysis of the SOL instabilities, showing good quantitative agreement.