  We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with
cubic anisotropy. We compute and analyze the fixed-dimension perturbative
expansion of the renormalization-group functions to four loops. The relations
of these models with N-color Ashkin-Teller models, discrete cubic models,
planar model with fourth order anisotropy, and structural phase transition in
adsorbed monolayers are discussed. Our results for N=2 (XY model with cubic
anisotropy) are compatible with the existence of a line of fixed points joining
the Ising and the O(2) fixed points. Along this line the exponent $\eta$ has
the constant value 1/4, while the exponent $\nu$ runs in a continuous and
monotonic way from 1 to $\infty$ (from Ising to O(2)). For N\geq 3 we find a
cubic fixed point in the region $u, v \geq 0$, which is marginally stable or
unstable according to the sign of the perturbation. For the physical relevant
case of N=3 we find the exponents $\eta=0.17(8)$ and $\nu=1.3(3)$ at the cubic
transition.
