Recently some new examples of domains in Euclidean and Riemannian spaces have been proved to satisfy P´olya conjecture for Dirichlet and/or Neumann eigenvalues. In a joint work with Pedro Freitas and Jing Mao (2022, to appear in Ann. Inst. Fourier) we show P´olya’s conjecture holds for n-dimensional hemispheres in the Neumann case, but not in the Dirichlet case when n ≥ 3, not even eventually. We derive P´olya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues and obtain direct and reversed Li-Yau inequalities for 2-sphere and 4-sphere, respectively. In another work with Freitas (JMP 2023), we show for instance that families of sectors of domains of revolution or geodesic disks, and thin cylinders satisfy P´olya’s conjecture. These examples are subdomains of a domain that satisfy P´olya conjecture eventually and can be partitioned by an arbitrarily large number of isometric copies of a subdomain of certain type. Domains having a second term in the Weyl asymptotic with the convenient sign satisfy P´olya conjecture eventually. We also improve the Li-Yau constant for general cylinders in the Dirichlet case.