  The critical thermodynamics of the two-dimensional N-vector cubic and MN
models is studied within the field-theoretical renormalization-group (RG)
approach. The beta functions and critical exponents are calculated in the
five-loop approximation and the RG series obtained are resummed using the
Borel-Leroy transformation combined with the generalized Pad\'e approximant and
conformal mapping techniques. For the cubic model, the RG flows for various N
are investigated. For N=2 it is found that the continuous line of fixed points
running from the XY fixed point to the Ising one is well reproduced by the
resummed RG series and an account for the five-loop terms makes the lines of
zeros of both beta functions closer to each another. For the cubic model with
N\geq 3, the five-loop contributions are shown to shift the cubic fixed point,
given by the four-loop approximation, towards the Ising fixed point. This
confirms the idea that the existence of the cubic fixed point in two dimensions
under N>2 is an artifact of the perturbative analysis. For the quenched dilute
O(M) models ($MN$ models with N=0) the results are compatible with a stable
pure fixed point for M\geq1. For the MN model with M,N\geq2 all the
non-perturbative results are reproduced. In addition a new stable fixed point
is found for moderate values of M and N.
