We enumerate central and medial quasigroups of order less than \(128\) up to isomorphism, with the exception of those quasigroups that are isotopic to \(C_4\times C_2^4\), \(C_2^6\), \(C_3^4\) or \(C_5^3\). We give an explicit formula for the number of quasigroups that are affine over a finite cyclic group.