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In this paper, we study the derived categories of a Koszul algebra and its Yoneda algebra to determine when those categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We have considered discrete Koszul algebras and we gave necessary and sufficient conditions for those Koszul algebras to be derived equivalent to their Yoneda algebras. We also study the class of Koszul algebras which are derived equivalent to hereditary algebras. For the case where the hereditary algebra is tame, we characterized the derived equivalence between those Koszul algebras and their Yoneda algebras [ABSTRACT FROM AUTHOR]