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In this thesis, some novel methods for solution of compressible, multi-phase flows on unstructured grids were developed. The developed methods are especially advantageous for interface problems, while they are also applicable to multi-phase flows containing mixtures as well as particle suspensions. The first method studied was a multi-dimensional, multi-phase Godunov method for compressible multi-phase flows. This method is based on the solution of a hyperbolic equation system for compressible multi-phase flows. There are several difficulties with this hyperbolic equation system due to non-conservative volume fraction equation and non-conservative terms also known as throttling therms existing in momentum and energy equations. Robust and accurate multi-dimensional discretization of these terms were derived based on Abgrall \cite{Abgrall} criterion. Next a new method based on discrete equations for multi-dimensional and multiphase problems on unstructured grids was developed. This method resolves all the problems associated with the non-conservative equations and terms. The high artificial numerical mixing of phase interfaces associated with available compressible schemes was resolved with a novel volume fraction differencing scheme. The developed differencing scheme used for volume fraction is the only scheme providing comparable resolution of the interfaces with tracking methods on multi-dimensional unstructured grids and very robust compared to other interface capturing methods studied in the related literature. The resulting methods provide ignorable numerical mixing of phase interfaces on ustructured solution grids while giving physically correct results for pressure and energy in contrast to other methods available in the literature. In addition to these solution methods, some special boundary conditions and preconditioning methods for low speed steady flows were applied. For high spatial resolution, combinations of linear reconstruction and Weighted Average Flux (WAF) methods were also applied in some problems.Ph.D. - Doctoral Progra