  Infrared and collinear safe event shape distributions and their mean values
are determined in e+e- collisions at centre-of-mass energies between 45 and 202
GeV. A phenomenological analysis based on power correction models including
hadron mass effects for both differential distributions and mean values is
presented. Using power corrections, alpha_s is extracted from the mean values
and shapes. In an alternative approach, renormalisation group invariance (RGI)
is used as an explicit constraint, leading to a consistent description of mean
values without the need for sizeable power corrections. The QCD beta-function
is precisely measured using this approach. From the DELPHI data on Thrust,
including data from low energy experiments, one finds beta_0 = 7.86 +/- 0.32
for the one loop coefficient of the beta-function or, assuming QCD, n_f = 4.75
+/- 0.44 for the number of active flavours. These values agree well with the
QCD expectation of beta_0=7.67 and n_f=5. A direct measurement of the full
logarithmic energy slope excludes light gluinos with a mass below 5 GeV.
