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We show that if $V$ is a vertex operator algebra such that all theirreducible ordinary $V$-modules are $C_1$-cofinite and all thegrading-restricted generalized Verma modules for $V$ are of finite length, thenthe category of finite length generalized $V$-modules has a braided tensorcategory structure.  By applying the general theorem to the simple affine vertex operator algebra(resp. superalgebra) associated to a finite simple Lie algebra (resp. Liesuperalgebra) $\mathfrak{g}$ at level $k$ and the category $KL_k(\mathfrak{g})$of its finite length generalized modules, we discover several families of$KL_k(\mathfrak{g})$ at non-admissible levels $k$, having braided tensorcategory structures. In particular, $KL_k(\mathfrak{g})$ has a braided tensorcategory structure if the category of ordinary modules is semisimple or moregenerally if the category of ordinary modules is of finite length.  We also prove the rigidity and determine the fusion rules of some categories$KL_k(\mathfrak{g})$, including the category $KL_{-1}(\mathfrak{sl}_n)$. Usingthese results, we construct a rigid tensor category structure on a fullsubcategory of $KL_1(\mathfrak{sl}(n|m))$ consisting of objects with semisimpleCartan subalgebra actions.

Tensor categories of affine Lie algebras beyond admissible levels