We initiate the study of computing diverse triangulations of a given polygon. Given a simple n-gon P, an integer \( k \ge 2 \) , a quality measure \(\sigma \) on the set of triangulations of P and a factor \( \alpha \ge 1 \) , we formulate the Diverse and Nice Triangulations (DNT) problem that asks to compute k distinct triangulations \(T_1,\dots ,T_k\) of P such that a) their diversity, \(\sum _{i < j} d(T_i,T_j) \) , is as large as possible and b) they are nice, i.e., \(\sigma (T_i) \le \alpha \sigma ^* \) for all \(1\le i \le k\) . Here, d denotes the symmetric difference of edge sets of two triangulations, and \(\sigma ^*\) denotes the best quality of triangulations of P, e.g., the minimum Euclidean length. As our main result, we provide a \(\textrm{poly}(n,k)\) -time approximation algorithm for the DNT problem that returns a collection of k distinct triangulations whose diversity is at least \(1 - \varTheta (1/k)\) of the optimal, and each triangulation satisfies the quality constraint. This is accomplished by studying bi-criteria triangulations (BCT), which are triangulations that simultaneously optimize two criteria, a topic of independent interest. We complement our approximation algorithms by showing that the DNT problem and the BCT problem are NP-hard. Finally, for the version where diversity is defined as \(\min _{i < j} d(T_i,T_j) \) , we show a reduction from the problem of computing optimal Hamming codes, and provide an \(n^{O(k)}\) -time \(\tfrac{1}{2}\) -approximation algorithm. This improves over the naive \({C_{n-2} \atopwithdelims ()k} \approx 2^{O(nk)}\) time bound for enumerating all k-tuples among the triangulations of a simple n-gon, where \(C_n\) denotes the n-th Catalan number.