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We develop a functorial approach to the study of the homotopy groups of spheres and Moore spaces $M(A,n)$, based on the Curtis spectral sequence and the decomposition of Lie functors as iterates of simpler functors such as the symmetric or exterior algebra functors. The discussion takes place over the integers, and includes a functorial description of the derived functors of certain Lie algebra functors, as well as of all the main cubical functors (such as the degree 3 component $SP^3$ of the symmetric algebra functor). As an illustration of this method, we retrieve in a purely algebraic manner the 3-torsion component of the homotopy groups of the 2-sphere up to degree 14, and give a unified presentation of homotopy groups $\pi_i(M(A,n))$ for small values of both $n$ and $i$., Comment: The text has been corrected and augmented. The former appendix A has been omitted, and will be posted separately