  The influence of non-magnetic impurities on the spectrum and dynamical spin
structure factor of a model for CuGeO$_3$ is studied. A simple extension to
Zn-doped ${\rm Sr Cu_2 O_3}$ is also discussed. Using Exact Diagonalization
techniques and intuitive arguments we show that Zn-doping introduces states in
the Spin-Peierls gap of CuGeO$_3$. This effect can beunderstood easily in the
large dimerization limit where doping by Zn creates ``loose'' S=1/2 spins,
which interact with each other through very weak effective antiferromagnetic
couplings. When the dimerization is small, a similar effect is observed but now
with the free S=1/2 spins being the resulting S=1/2 ground state of severed
chains with an odd number of sites. Experimental consequences of these results
are discussed. It is interesting to observe that the spin correlations along
the chains are enhanced by Zn-doping according to the numerical data presented
here. As recent numerical calculations have shown, similar arguments apply to
ladders with non-magnetic impurities simply replacing the tendency to
dimerization in CuGeO$_3$ by the tendency to form spin-singlets along the rungs
in SrCu$_2$O$_3$.
