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 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 obtains
beta_0 = 7.86 +- 0.32
for the one loop coefficient of the beta-function or, assuming QCD for the number of active flavours,
n_f = 4.75 +- 0.44
These values agree well with the QCD expectation of beta_0=7.67 and n_f=5.