chapter 1 Linear Invariant Subspaces in Quasilinear Equations: Basic Examples and Models -- chapter 2 Invariant Subspaces and Modules: Mathematics in One Dimension -- chapter 3 Parabolic Equations in One Dimension: Thin Film, Kuramoto-Sivashinsky, and Magma Models -- chapter 4 Odd-Order One-Dimensional Equations: Korteweg-de Vries, Compacton, Nonlinear Dispersion, and Harry Dym Models -- chapter 5 Quasilinear Wave and Boussinesq Models in One Dimension. Systems of Nonlinear Equations -- chapter 6 Applications to Nonlinear Partial Differential Equations in IR N -- chapter 7 Partially Invariant Subspaces, Invariant Sets, and Generalized Separation of Variables -- chapter 8 Sign-Invariants for Second-Order Parabolic Equations and Exact So- lutions -- chapter 9 Invariant Subspaces for Discrete Operators, Moving Mesh Methods, and Lattices.