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We review some recent results obtained in the framework of the 2-dimensionalInteracting Self-Avoiding Walk (ISAW). After a brief presentation of therigorous results that have been obtained so far for ISAW we focus on theInteracting Partially Directed Self-Avoiding Walk (IPDSAW), a model introducedin Zwanzig and Lauritzen (1968) to decrease the mathematical complexity ofISAW. In the first part of the paper, we discuss how a new probabilisticapproach based on a random walk representation (see Nguyen and P\'etr\'elis(2013)) allowed for a sharp determination of the asymptotics of the free energyclose to criticality (see Carmona, Nguyen and P\'etr\'elis (2016)). Somescaling limits of IPDSAW were conjectured in the physics literature (see e.g.Brak et al. (1993)). We discuss here the fact that all limits are now provenrigorously, i.e., for the extended regime in Carmona and P\'etr\'elis (2016),for the collapsed regime in Carmona, Nguyen and P\'etr\'elis (2016) and atcriticality in Carmona and P\'etr\'elis (2017a). The second part of the paperstarts with the description of four open questions related to physicallyrelevant extensions of IPDSAW. Among such extensions is the Interacting PrudentSelf-Avoiding Walk (IPSAW) whose configurations are those of the 2-dimensionalprudent walk. We discuss the main results obtained in P\'etr\'elis and Torri(2016+) about IPSAW and in particular the fact that its collapse transition isproven to exist rigorously.

Interacting partially directed self-avoiding walk: a probabilistic perspective