  This work is a contribution to the study of universality in
out-of-equilibrium lattice models undergoing a second-order phase transition at
equilibrium. The experimental protocol that we have chosen is the following:
the system is prepared in its high-temperature phase and then quenched at the
critical temperature $T_c$. We investigated by mean of Monte Carlo simulations
two quantities that are believed to take universal values: the exponent
$\lambda/z$ obtained from the decay of autocorrelation functions and the
asymptotic value $X_\infty$ of the fluctuation-dissipation ratio $X(t,s)$. This
protocol was applied to the Ising model, the 3-state clock model and the
4-state Potts model on square, triangular and honeycomb lattices and to the
Ashkin-Teller model at the point belonging at equilibrium to the 3-state Potts
model universality class and to a multispin Ising model and the Baxter-Wu model
both belonging to the 4-state Potts model universality class at equilibrium.
