Using a sample of 69,919 red giants from the SDSS-III/APOGEE Data Release 12, we measure the distribution of stars in the [$alpha$/Fe] vs. [Fe/H] plane and the metallicity distribution functions (MDF) across an unprecedented volume of the Milky Way disk, with radius $3<R<15$ kpc and height $|z|<2$ kpc. Stars in the inner disk ($R<5$ kpc) lie along a single track in [$alpha$/Fe] vs. [Fe/H], starting with $alpha$-enhanced, metal-poor stars and ending at [$alpha$/Fe]$sim0$ and [Fe/H]$sim+0.4$. At larger radii we find two distinct sequences in [$alpha$/Fe] vs. [Fe/H] space, with a roughly solar-$alpha$ sequence that spans a decade in metallicity and a high-$alpha$ sequence that merges with the low-$alpha$ sequence at super-solar [Fe/H]. The location of the high-$alpha$ sequence is nearly constant across the disk, however there are very few high-$alpha$ stars at $R>11$ kpc. The peak of the midplane MDF shifts to lower metallicity at larger $R$, reflecting the Galactic metallicity gradient. Most strikingly, the shape of the midplane MDF changes systematically with radius, with a negatively skewed distribution at $3<R<7$ kpc, to a roughly Gaussian distribution at the solar annulus, to a positively skewed shape in the outer Galaxy. For stars with $|z|>1$ kpc or [$alpha$/Fe]$>0.18$, the MDF shows little dependence on $R$. The positive skewness of the outer disk MDF may be a signature of radial migration; we show that blurring of stellar populations by orbital eccentricities is not enough to explain the reversal of MDF shape but a simple model of radial migration can do so.
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