  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 $i$.
