This thesis deals with Opial inequality and its famous generalizations, extensions and refinements, i.e. with Opial-type inequalities. First, new Opial-type inequalities for convex functions are presented, some of which are generalization and some are refinement of Opial’s, Willett’s, Godunova-Levin’s and Rozanova’s inequality. Then results for more variables are proven. Moreover, new Opial-type inequalites are obtained for convex and relatively convex functions closelly connected to Mitrinović-Pečarić’s results, and application of these results yields inequalities for fractional derivatives of the Riemann-Liouville, Canavati and Caputo type and for the Riemann-Liouville fractional integrals. Mean value results are proven for linear functionals constructed from the newly obtained inequalities. Several examples of families of functions are given which enable construction of families of exponentially convex function and monotonic Stolarsky-type means. Further, general Opial-type inequalities for measurable functions and for the quotient of functions are obtained. Application to numerous symmetric functions gives results that involve Green’s functions, Lidstone’s series and Hermite’s interpolating polynomials.