We consider problems formulated and solved in terms of tropical mathematics to minimize functions defined on vectors over semifields with idempotent addition and invertible multiplication, subject to constraints given by vector equations and inequalities. We start with an overview of known tropical optimization problems and available solution methods. Then, we examine new problems and give direct solutions in a compact vector form ready for practical implementation. Finally, we apply the results obtained to solve real world problems in project scheduling, location analysis and decision making.