Lepton Flavour Violating (LFV) $\tau$ lepton decays are strongly suppressed in the Standard Model but some New Physics models predict the branching fractions of decays such as $\tau \to \ell \ell \ell$ and $\tau \to \mu \nu$ in the experimentally accesible range of $10^{-9}$ to $10^{-7}$. Belle has studied 48 different LFV $\tau$ lepton modes and set 90% C.L. upper limits of $\mathcal{O}(10^{-8})$ on the branching fractions. In this paper we review these studies and discuss prospects of LFV searches in $\tau$ lepton decays at future Belle II experiment which should collect 50 times higher statistics than Belle.