In the proposed study, we present several significant results annexed to the wellknown Hermite-Hadamard inequality. Also, we focus on various newly established classes of convex functions and their corresponding variants of Hadamard type inequalities. This PhD dissertation is devoted to sift out certain inequalities of Hadamard type from the class of convex functions to their recently established versions, namely MT-convex functions, co-ordinated convex functions etc. In addition, we are mainly concerned with various updated versions and analogues of the well-known Hermite-Hadamard inequalities in terms of integrations such as, classical integrals, Riemann-Liouville’s fractional integrals and α-fractional conformable integrals. Eventually, as applications, the proposed results are further utilized to achieve some novel bounds for special means of positive real numbers. Also, some explicit bounds are also being derived to the versatile composite quadrature rules in terms of distinct functions belonging to different classes of convex functions. At the end, different inequalities have been obtained pertaining to F-divergence measures. In the first chapter, we present some basic concepts, certain necessary terminologies and recall a few important results from the theory of convex analysis in general, and convex sets and convex functions in particular, where many of them will be encountered through out the thesis. Also, these core and elementary notions will provide comparatively a better foundation to the readers in the understanding of the proposed study. In the second chapter, we present several integral identities for differentiable, twice differentiable and three times differentiable functions connected with both left and right hand parts of the classical Hermite-Hadamard inequality. Then, we obtain various Hadamard type inequalities based on these identities via classical integrals. These results have some natural applications to special means of real numbers and trapezoidal as well as midpoint formulas. In the third chapter, we discover two novel integral identities for twice differentiable functions. Then, we employ these identities to establish some general inequalities for the functions whose second derivatives absolute values are MTconvex. These inequalities provide us some new estimates for the right hand side of the Hermite-Hadamard type inequalities for classical integrals and Riemann- Liouville’s fractional integrals. Next, by making use of these results, we point out applications to some means of real numbers and several error estimations for the trapezoidal formula. In the fourth chapter, we obtain some new Hermite-Hadamard type inequalities for convex functions on the co-ordinates. These results refine the earlier work done by Dragomir and Chen . In the fifth chapter, we establish two integral identities for conformable fractional integrals. Then, under the utility these results, we design several integral inequalities connected with the left and right hand side of the Hermite-Hadamard type inequalities for conformable fractional integrals. These results extends the earlier known results from classical integrals to conformable fractional integrals. In the sixth chapter, we give applications of our main results established in the Chapters 2, 3, 4 and 6 respectively. In addition to that, in Section 6.1 applications to special means of real number are provided. Then, in the next Section 6.2, some new error estimates for trapezoidal formula are given. Furthermore, in the next Section 6.3, error estimates for midpoint formula are addressed. In the last Section 6.4, some applications to F-Divergence measures are provided.