This article suggests a series of problems related to various algebraic and geometric aspects of integrability of finite-dimensional Hamiltonian systems. They reflect some recent developments in the following research directions: existence of integrable Hamiltonian systems on Poisson and symplectic manifolds; bi-Poisson geometry and the argument-shift method in relation to the Mischenko-Fomenko conjecture; different types of Poisson pencils according to the Jordan-Kronecker decomposition; the Jordan-Kronecker invariants of finite-dimensional Lie algebras and relation of flatness of the pencil to completeness of the ring of polynomial invariants; quantization of the argument-shift method; applications to differential geometry, including sectional operators, projective equivalence and holonomy groups; interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, including singularities of bi-Hamiltonian systems and bi-Hamiltonian reduction. The authors recall the state of knowledge before stating the problems.