1 Basic results for linear equations -- 1.1 Linear differential equations and linear algebraic equations -- 1.2 Exponential dichotomy and L2evolutions -- 2 Hamiltonian Matrices and Algebraic Riccati equations -- 2.1 Solutions of algebraic Riccati equations and graph subspaces -- 2.2 Indefinite scalar products and a canonical form of Hamiltonian matrices -- 2.3 Hermitian algebraic Riccati equations -- 2.4 Positive semi-definite solutions of standard algebraic Riccati equations -- 2.5 Hermitian discrete-time algebraic Riccati equations -- 3 Global aspects of Riccati differential and difference equations -- 3.1 Riccati differential equations and associated linear systems -- 3.2 A representation formula -- 3.3 Flows on Graßmann manifolds: The extended Riccati differential equation -- 3.4 General representation formulae for solutions of RDE and PRDE, the time-continuous and periodic Riccati differential equation, and dichotomy -- 3.5 A representation formula for solutions of the discrete time Riccati equation -- 3.6 Global existence results -- 4 Hermitian Riccati differential equations -- 4.1 Comparison results for HRDE -- 4.2 Monotonicity and convexity results: A Fréchet derivative based approach -- 4.3 Convergence to the semi-stabilizing solution -- 4.4 Dependence of HRDE on a parameter -- 4.5 An existence theorem for general HRDE -- 4.6 A special property of HRDE -- 5 The periodic Riccati equation -- 5.1 Linear periodic differential equations -- 5.2 Preliminary notation and results for linear periodic systems -- 5.3 Existence results for periodic Hermitian Riccati equations -- 5.4 Positive semi-definite periodic equilibria of PRDE -- 6 Coupled and generalized Riccati equations -- 6.1 Some basic concepts in dynamic games -- 6.2 Non-symmetric Riccati equations in open loop Nash differential games -- 6.3 Discrete-time open loop Nash Riccati equations -- 6.4 Non-symmetric Riccati equations in open loop Stackelberg differential games -- 6.5 Discrete-time open loop Stackelberg equations -- 6.6 Coupled Riccati equations in closed loop Nash differential games -- 6.7 Rational matrix differential equations arising in stochastic control -- 6.8 Rational matrix difference equations arising in stochastic control -- 6.9 Coupled Riccati equations in Markovian jump systems -- 7 Symmetric differential Riccati equations: an operator based approach -- 7.1 Popov triplets: definition and equivalence -- 7.2 Associated objects -- 7.3 Associated operators -- 7.4 Existence of the stabilizing solution -- 7.5 Positivity theory and applications -- 7.6 Differential Riccati inequalities -- 7.7 The signature condition -- 7.8 Differential Riccati theory: A Hamiltonian descriptor operator approach -- 8 Applications to Robust Control Systems -- 8.1 The Four Block Nehari Problem -- 8.2 Disturbance Attenuation -- 9 Non-symmetric Riccati theory and applications -- 9.1 Non-symmetric Riccati theory -- 9.2 Application to open loop Nash games -- 9.3 Application to open loop Stackelberg games -- A Appendix -- A.1 Basic facts from control theory -- A.2 The implicit function theorem -- References -- List of Figures.