Using a Monte-Carlo method, we study the behaviour of different lattice models of a linear macromolecule in the theta-point region. Paying most attention to the testing of new theoretical concepts of the two-dimensional theta-point 1,2, we give a new interpretation of some numerical results of our previous work 3,4 and generalize others. We confirm the assumption of the universality of the value of the tricritical exponent nu(t) = 4/7. The values of nu(t) for a self-avoiding walk (SAW) on triangular, square and honeycomb lattices as well as for two special models, viz. infinitely prolonging self-avoiding walks (IPSAW) 3 and infinitely growing self-avoiding walks with nearest-neighbour interactions (IGSAWN) 5, lie between 0.56 and 0.59. The slight differences between these results can be explained by the effect of the corrections to scaling, which have different values for different models. The values of the crossover exponent phi(t) are confined between 0.42 and 0.6, which is not in complete contradiction with the value phi(t) = 3/7 for the theta-point, proposed elsewhere 2, but differ from our previous result phi(t) = 0.6 +/- 0.1 obtained earlier 3,4 without extrapolation to the limit N --> infinity, where N is the length of a chain. The values of the free-energy exponent gamma(t) determined in the present work lie between 0.98 and 1.07, and, thus, do not coincide with the value of gamma(t) = 8/7 proposed by others 2. The value of gamma-1t = 0.5 +/- 0.05 that we obtained for a polymer confined in a half-space is in complete contradiction with the above result 2 for the theta-point. As a whole, our results are in good agreement with recent work 6,7 using a similar technique for shorter SAWs on a square lattice.