In this monograph, we established certain new integral inequalities for n-time differentiable mappings along with some new (H-H)'s type integral inequalities for mappings of several variables for different kinds of convexities. Also, we generalized results that are connected with celebrated 's integral inequality with different kinds of convexities. Some parallel developments are formulated which based on concavity. Some applications for special means of real numbers are also provided which can naturally be deduced. Some particular inequalities are investigated which include explicit bounds for trapezoid, midpoint, Simpson's, Newton-Cotes and left and right rectangle rules.