Authors’ abstract: Orientable triangular embeddings of the complete tripartite graph \(K_{n,n,n}\) correspond to biembeddings of Latin squares. We show that if \(n\) is prime there are at least \(e^{n\ln n-n(1+ o(1))}\) nonisomorphic biembeddings of cyclic Latin squares of order \(n\). If \(n= kp\), where \(p\) is a large prime number, then the number of nonisomorphic biembeddings of cyclic Latin squares of order \(n\) is at least \(e^{p\ln p-p(1+ \ln k+ o(1))}\). Moreover, we prove that for every \(n\) there is a unique regular triangular embedding of \(K_{n,n,n}\) in an orientable surface.