The talk is about a class of systems of 2d statistical mechanics, such as random tilings, noncolliding walks, log-gases and random matrix-type distributions. Specific members in this class are integrable, which means that available exact formulas allow delicate asymptotic analysis leading to the Gaussian Free Field, sine-process, Tracy-Widom distributions. Extending the results beyond the integrable cases is challenging. I will speak about a recent progress in this direction: about universal local limit theorems for a class of lozenge and domino tilings, noncolliding random walks; and about GFF-type asymptotic theorems for global fluctuations in these systems and in discrete beta log-gases.