  We extent the standard approach of dimensional regularization of Feynman
diagrams: we replace the transition to lower dimensions by a 'natural' cut-off
regulator. Introducing an external regulator of mass Lambda^(2e), we regain in
the limit e -> 0 and e > 0 the results of dimensional and cut-off
regularization, respectively. We demonstrate the versatility and adequacy of
the different regularization schemes for practical examples (such as non
covariant regularization, the axial anomaly or regularization in effective
field theories).
