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Based on new and original results obtained by the authors, in collaboration with other leading researchers such as R Deeley, J T Horwood, R Milson and Jin Yue, this book presents a new invariant theory along with many interesting applications. A central theme of the book is that ITKT (invariant theory of Killing tensors) is a pseudo-Riemannian analogue of the classical invariant theory. The book presents a self-contained review obtained so far in the development of ITKT. It also provides a new approach to the study of Hamiltonian systems of classical mechanics. Although the subject lies at the intersection of differential geometry, Lie group theory, classical mechanics and invariant theory, the book is self-contained and accessible to upper years undergraduate and graduate students.