  We study different notions of Riemannian curvatures: The $p$-curvatures which
interpolate between the scalar curvature and the sectional curvature, the
Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar
curvature to the Gauss-Bonnet integrand. We bring out the $(p,q)$-curvatures,
which incorporate all the previous curvatures. We then examine the curvature
term which appears in the classical Weitzenb\"ock formula. We also study the
positivity properties of the $p$-curvatures, the second Gauss-Bonnet-Weyl
curvature, the Einstein curvature and the isotropic curvature.
