  The early stages of a galaxy's evolution leave an imprint on its metallicity
distribution. We discuss the origins and evolution of radial metallicity
gradients in discs of spiral galaxies using an analytical chemical evolution
model. We explain how radial metallicity gradients in stellar populations are
determined by three factors: the radial metallicity profile of the star-forming
ISM, radial changes in the star-formation history (in particular inside-out
formation), and radial mixing of stars. Under reasonable assumptions,
inside-out formation steepens the negative ISM metallicity gradient, but
contributes positively to the stellar metallicity gradient, up to inverting the
metallicity profile to a positive d[Fe/H]/dR. This reconciles steep negative
d[Fe/H]/dR in some high redshift galaxies to generally flatter gradients in
local observations.
  We discuss the evidence for inverse radial metallicity gradients (positive
d[X/H]/dR) at high redshifts and the inverse relationship between azimuthal
velocity and the metallicity (positive dV_{\phi}/d[Fe/H]) of stars for the
Milky Way's thick disc. The former can be achieved by high central gas-loss
rates and re-distribution processes, e.g. re-accretion of enriched material in
conjunction with inside-out formation, and near-disc galactic fountaining. For
the Milky Way thick disc, we show that the positive dV_{\phi}/d[Fe/H]
correlation points to comparable timescales for inside-out formation, initial
metal enrichment and SNIa enrichment. We argue that the original ISM
metallicity gradient could be inferred with better data from the
high-metallicity tail of the alpha enhanced population. Including inside-out
formation in our models changes the local vertical metallicity gradient by
about -0.2dex/kpc, in line with local measurements.
