In this talk,  we will discuss the following problem:  Given a bounded domain $\Omega$ in R^n, and a positive energy N, one divides $\Omega$ into N subdomains, $\Omega_j, j= 1, 2,..., N$. We consider the so-called optimal partitions that give the least possible value for the sum of the first Dirichelet eigenvalues on these sumdomains among all admissible partitions of $\Omega$. For given N the problem has been studied by various authors. I shall discuss some recent progress and conjectures on the analysis on asymptotic behavior these optimal partitions as N tends to infinite.