First we survey known results on spectral geometry of K\"{a}hler manifolds. Then we study the question of the spectral characterization of $\mathbb{CP}^n$. Namely for each fixed nonnegative integers $p$, if a compact Kähler manifold M of complex dimension $n$ has the same $p$-spectra as $\mathbb{CP}^n$ equipped withthe Fubini-Study metric, we give explicit range of $n$ such that this Kähler manifold is holomorphically isometric to $\mathbb{CP}^n$. This extends previous works of Tanno, Chen-Vanhecke, Goldberg for $p\leq 2$ and Ping Li for even $p$. This is joint work with K. Liu, X. Huang and Y. Zhi.