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We have studied the properties of Seyfert galaxies with high [OIII]5007 blueshifts (`blue outliers'), originally identified because of their strong deviation from the M_BH - sigma relation of normal, narrow-line Seyfert 1 (NLS1) and broad-line Seyfert 1 (BLS1) galaxies. These blue outliers turn out to be important test-beds for models of the narrow-line region (NLR), for mechanisms of driving large-scale outflows, for links between NLS1 galaxies and radio galaxies, and for orientation-dependent NLS1 models. We report the detection of a strong correlation of line blueshift with ionization potential in each galaxy, including the measurement of coronal lines with radial velocities up to 500--1000 km/s. All [OIII] blue outliers have narrow widths of their broad Balmer lines and high Eddington ratios. While the presence of non-shifted low-ionization lines signifies the presence of a classical outer quiescent NLR in blue outliers, we also report the absence of any second, non-blueshifted [OIII] component from a classical inner NLR. These results place tight constraints on NLR models. We favor a scenario in which the NLR clouds are entrained in a decelerating wind which explains the strong stratification and the absence of a zero-blueshift inner NLR of blue outliers. The origin of the wind remains speculative at this time (collimated radio plasma, thermal winds, radiatively accelerated clouds). It is perhaps linked to the high Eddington ratios of blue outliers. Similar, less powerful winds could be present in all Seyfert galaxies, but would generally only affect the coronal line region (CLR), or level off even before reaching the CLR. Similarities between blue outliers in NLS1 galaxies and (compact) radio sources are briefly discussed.