  We define a geometrically meaningful compactification of the moduli space of
smooth plane curves, which can be calculated explicitly. The basic idea is to
regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a
divisor, and allow both the surface and the curve to degenerate. For plane
curves of degree d at least 4, we obtain a compactification M_d which is a
moduli space of stable pairs (X,D) using the log minimal model program. A
stable pair (X,D) consists of a surface X such that -K_X is ample and a divisor
D in a given linear system on X with specified singularities. Note that X may
be non-normal, and K_X is Q-Cartier but not Cartier in general. We give a rough
classification of stable pairs of arbitrary degree, a complete classification
in degrees 4 and 5, and a partial classification in degree 6. The
compactification is particularly simple if d is not a multiple of 3 - in
particular the surface X has at most 2 components. We give a characterisation
of these surfaces in terms of the singularities and the Picard numbers of the
components. Moreover, we show that M_d is smooth in this case.
