A detailed description of inequalities, their types, significance, and their applications in various fields of Mathematics & Engineering. Many classical inequalities are been discussed in this book; relationships among these inequalities and the usage of these specific inequalities is been discussed. Gronwall-Bellman inequality is one of the important inequality present in literature which was later considered & generalized by many authors in literature. Few new results of Gronwall-Bellman type inequality which are been published in my recent paper are also been elaborated with detail in this book. Along with it, applications of these newly generated results can also be seen in this book, which shows the validity of those newly generated results based on generalized Gronwall-Bellman type inequalities.