  The complete classification of WZNW modular invariant partition functions is
known for very few affine algebras and levels, the most significant being all
levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address
the classification problem for the nicest high rank semi-simple affine
algebras: $(A_1^{(1)})^{\oplus_r}$. Among other things, we explicitly find all
automorphism invariants, for all levels $k=(k_1,\ldots,k_r)$, and complete the
classification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. We
also solve the classification problem for $(A_1^{(1)})^{\oplus_r}$, for any
levels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\leq 3$.
In addition, we find some physical invariants which seem to be new. Together
with some recent work by Stanev, the classification for all
$(A^{(1)}_1)^{\oplus_r}_k$ could now be within sight.
