  In this paper, we study the Cauchy and Goursat problems of the spin-$n/2$
zero rest-mass equations on Minkowski spacetime by using the conformal
geometric method. In our strategy, we prove the wellposedness of the Cauchy
problem in Einstein's cylinder. Then we establish pointwise decays of the
fields and prove the energy equalities of the conformal fields between the null
conformal boundaries $\scri^\pm$ and the hypersurface $\Sigma_0=\left\{ t=0
\right\}$. Finally, we prove the wellposedness of the Goursat problem in the
partial conformal compactification by using the energy equalities and the
generalisation of H\"ormander's result.
