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In this paper, we report on the $\tau$-tilting finiteness of some classes offinite-dimensional algebras over an algebraically closed field, includingsymmetric algebras of polynomial growth, $0$-Hecke algebras, and $0$-Schuralgebras. Consequently, we find that derived equivalence preserves the$\tau$-tilting finiteness over symmetric algebras of polynomial growth, andself-injective cellular algebras of polynomial growth are $\tau$-tiltingfinite. Furthermore, the representation-finiteness and $\tau$-tiltingfiniteness over 0-Hecke algebras and 0-Schur algebras (with few exceptions)coincide.

On $\tau$-tilting finiteness of symmetric algebras of polynomial growth, $0$-Hecke and $0$-Schur algebras