  The celebrated P\'{o}lya's conjecture (1954) in spectral geometry states that
the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a
bounded Euclidean domain can be estimated from above and below, respectively,
by the leading term of Weyl's asymptotics. P\'{o}lya's conjecture is known to
be true for domains which tile Euclidean space, and, in addition, for some
special domains in higher dimensions. In this paper, we prove P\'{o}lya's
conjecture for the disk, making it the first non-tiling planar domain for which
the conjecture is verified. We also confirm P\'{o}lya's conjecture for
arbitrary planar sectors, and, in the Dirichlet case, for balls of any
dimension. Along the way, we develop the known links between the spectral
problems in the disk and certain lattice counting problems. A key novel
ingredient is the observation, made in recent work of the last named author,
that the corresponding eigenvalue and lattice counting functions are related
not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs
are purely analytic, except for a rigorous computer-assisted argument needed to
cover the short interval of values of the spectral parameter in the case of the
Neumann problem in the disk.
