Multidimensional optimization problems in the tropical mathematics setting are considered. The problems are to minimize (maximize) linear and nonlinear functionals defined on finite-dimensional semimodules over idempotent semifields, subject to constraints in the form of linear equations and inequalities. We outline known problems and discuss their solution methods. New unconstrained and constrained optimization problems are then examined and related exact solutions are given in a compact vector form. As an application, we present solutions of real-world problems in project scheduling and location analysis.