The spin-spin correlation function in the ground state of some two-dimensional fully frustrated spin models are investigated. The correlations are long ranged and decay as $\frac{1}{{r}^{\ensuremath{\eta}}}$, where
r is the distance between the two spins and $\ensuremath{\eta}=\frac{1}{2}$. This result is true for both the square and triangular fully frustrated lattices, thus suggesting that these models form a universality class. It is shown that the fully frustrated Ising model on a square lattice can be mapped exactly into a special Baxter model. The spin-spin correlation function of this Baxter model can be calculated exactly. The ground state of the fully frustrated square Ising model corresponds to the free fermion point of the
F model and therefore, through duality, to the decoupling point of the critical Baxter line. The relationship between the
F model above the critical temperature and the Gaussian model above the multicritical point is discussed.
The phase diagram of a two-dimensional frustrated Ising model with both anti-ferromagnetic and ferromagnetic couplings is studied using tensor network renormalization group techniques. This model can be seen as two antiferromagnetic Ising replicas coupled by nonlocal spin-spin interactions, designed in such a way that the continuum limit matches that of the still-debated J_{1}-J_{2} model and induces a marginal critical behavior. Our model has the advantage of having more symmetries than the J_{1}-J_{2} model and of allowing a more straightforward implementation of tensor network renormalization group algorithms. We demonstrate the existence of two transition lines, featuring both first- and second-order regimes. In the latter, the central charge and the critical exponents are shown to be compatible with the Ashkin-Teller universality class. This picture is consistent with that given by Monte Carlo simulations of the J_{1}-J_{2} model but not with recent studies with tensor network techniques.