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Nowadays Mathematical Inequality Theory is a very active and attractive field of mathematics and provides a wide-spread research area in many branches of mathematics. This monograph mainly studies some important inequalities such as Jensen, AGM, Ky Fan, Cauchy-Schwartz, Triangle, Bessel, Hermite-Hadamard and etc. In this work, we give some new proofs, refinements and extensions to these inequalities together with obtaining various variants to them. The main tools used are Convexity, Product Measure, Fubini''s theorem, Discretization, Compactization and Binomial Expansion. The audiences of this monograph are Ph.D. students and professors working in the field of Inequality Theory, and because of its fluency, it can be used by interested undergraduates for finding new results and examples in classical Mathematical Analysis.