  We study realizations of the exceptional non-linear (quadratically generated,
or W-type) N=8 and N=7 superconformal algebras with Spin(7) and G_2 affine
symmetry currents, respectively. Both the N=8 and N=7 algebras admit unitary
highest-weight representations in terms of a single boson and free fermions in
8 of Spin(7) and 7 of G_2, with the central charges c_8=26/5 and c_7=5,
respectively. Furthermore, we show that the general coset Ans"atze for the N=8
and N=7 algebras naturally lead to the coset spaces SO(8)xU(1)/SO(7) and
SO(7)xU(1)/G_2, respectively, as the additional consistent solutions for
certain values of the central charge. The coset space SO(8)/SO(7) is the
seven-sphere S^7, whereas the space SO(7)/G_2 represents the seven-sphere with
torsion, S^7_T. The division algebra of octonions and the associated triality
properties of SO(8) play an essential role in all these realizations. We also
comment on some possible applications of our results to string theory.
