This book gives an in-depth treatment of optical phase space and Lie-algebra based approaches to classical optics, providing a unifying context for ray and wave optics. As well as providing mathematical tools for those using classical optical techniques, the book describes the effectiveness of optics in providing a more tangible interpretation of abstract mathematical concepts. Chapter 1 introduces the basic mathematical constructs of Hamiltonian Optics and Optical Phase Space, which will be used throughout the book. In chapters 2 & 3, the Hamiltonian approach is extended to ray optics via the matrix formalism, and wave optics via the transfer operator formalism. More advanced concepts covered in later chapters include the Fractional Fourier Transform and the Wigner Distribution Function. Two appendices provide reviews of Lie Group theory and the paraxial wave equation.