MATHEMATICAL BACKGROUND; Sobolev Spaces; Set-Valued Analysis; Nonsmooth Analysis; Nonlinear Operators; Elliptic Differential Equations; Remarks; CRITICAL POINT THEORY; Locally Lipschitz Functionals; Constrained Locally Lipschitz Functionals; Perturbations of Locally Lipschitz Functionals; Local Linking and Extensions; Continuous Functionals; Multivalued Functionals; Remarks; ORDINARY DIFFERENTIAL EQUATIONS; Dirichlet Problems; Periodic Problems; Nonlinear Boundary Conditions; Variational Methods; Method of Upper and Lower Solutions; Positive Solutions and Other Methods; Hamiltonian Inclusions; Remarks; ELLIPTIC EQUATIONS; Problems at Resonance; Neumann Problems; Problems with an Area-Type Term; Strongly Nonlinear Problems; Method of Upper and Lower Solutions; Multiplicity Results; Positive Solutions; Problems with Discontinuous Nonlinearities; Remarks; APPENDIX; Set Theory and Topology; Measure Theory; Functional Analysis; Nonlinear Analysis; List of Symbols; References.