A topological approach to quantisation of closed surfaces is presented. This is done by imposing the following ``minimal'' requirements for a quantisation \(R\) of a closed Riemann surface \(\Sigma\) of genus \(g\geq 2\): \(R\) is a unital \(C^*\)-algebra, both \(R\) and \(C(\Sigma)\) are fibres in a continuous family of \(C^*\)-algebras over a path connected space, and \(R\) is \(KK\)-equivalent to a twisted \(C^*\)-algebra \(C^*_{\text{red}}(\Gamma,\sigma)\) where \(\Gamma\) is the fundamental group of \(\Sigma\) and \(\sigma\) is a group 2-cocycle. A family \(\{R_s\}_{s\in (2,\infty]}\) of \(C^*\)-algebras satisfying above requirements is constructed by using the fact that the universal covering space of \(\Sigma\) can be modelled by the Poincaré disc \(\mathbb D\), and \(\Gamma\) can be identified with a discret subgroup of \(PSU(1,1)\) acting on \(\mathbb D\). The \(K\)-theory and representation theory of \(R_s\) are also described.