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We study the geometry of 3-codimensional smooth subvarieties of the complex projective space. In particular, we classify all quasi-Buchsbaum Calabi-Yau threefolds in projective 6-space. Moreover, we prove that this classification includes all Calabi-Yau threefolds contained in a possibly singular 5-dimensional quadric as well as all Calabi-Yau threefolds of degree at most 14 in P6.