In this paper,we give a survey on the Hill-type formula and its applications.Moreover,we generalize the Hill-type formula for linear Hamiltonian systems and Sturm-Liouville systems with any self-adjoint boundary conditions,which include the standard Neumann,Dirichlet and periodic boundary conditions.The Hill-type for-mula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions.Further,based on the Hill-type formula,we derive the Krein-type trace formula.As applications,we give nontrivial estimations for the eigenvalue problem and the relative Morse index.