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In 1966, B. O'Neill [The fundamental equations of a submersion, Michigan Math. J., Volume 13, Issue 4 (1966), 459-469.] defined some fundamental equations and curvature rela- tions between the total space, the base space and the fibres on a submersion. In the present paper, we define new curvature tensors on Riemannian submersions such as Weyl projective curvature tensor, concircular curvature tensor, conharmonic curvature tensor, conformal curvature tensor and M projective curvature tensor, respectively. Finally, we obtain some results in case of the total space of Riemannian submersions has umbilical fibres for any curvature tensors mentioned by the above. [ABSTRACT FROM AUTHOR]