Transport analyses using first-principle turbulence codes and 11/2-D transport codes usually study radial transport properties between the tokamak plasma magnetic axis and a normalized minor radius around 0.8. In this region, heat transport shows significantly stiff properties resulting in temperature scalelength values (R/L-T) that are relatively independent of the level of the radial heat flux. We have studied experimentally in the tokamak a configuration variable [F. Hofmann et al., Plasma Phys. Controlled Fusion 36, B277 (1994)] the radial electron transport properties of the edge region, close to the last closed flux surface, namely, between rho(V) = root V/V-edge = 0.8 to 1. It is shown that electron transport is not stiff in this region and high R/L-Te values (similar to 20-40) can be attained even for L-mode confinement. We can define a "pedestal" location, already in L-mode regimes, where the transport characteristics change from constant logarithmic gradient, inside rho(V) = 0.8, to constant gradient between 0.8 and 1.0. In particular, we demonstrate, with well resolved T-e and n(e) profiles, that the confinement improvement with plasma current I-p, with or without auxiliary heating, is due to this non-stiff edge region. This new result is used to explain the significant confinement improvement observed with negative triangularity, which could not be explained by theory to date. Preliminary local gyrokinetic simulations are now consistent with an edge, less stiff, region that is more sensitive to triangularity than further inside. We also show that increasing the electron cyclotron heating power increases the edge temperature inverse scalelength, in contrast to the value in the main plasma region. The dependence of confinement on density in ohmic plasmas is also studied and brings new insight in the understanding of the transition between linear and saturated confinement regimes, as well as of the density limit and appearance of a 2/1 tearing mode. The results presented in this paper provide an important new perspective with regards to radial transport in tokamak plasmas which goes beyond L-mode plasmas and explains some previous puzzling results. It is proposed that understanding the transport properties in this edge non-stiff region will also help in understanding the improved and high confinement edge properties. (C) 2014 AIP Publishing LLC.