  We review some recent results obtained in the framework of the 2-dimensional
Interacting Self-Avoiding Walk (ISAW). After a brief presentation of the
rigorous results that have been obtained so far for ISAW we focus on the
Interacting Partially Directed Self-Avoiding Walk (IPDSAW), a model introduced
in Zwanzig and Lauritzen (1968) to decrease the mathematical complexity of
ISAW. In the first part of the paper, we discuss how a new probabilistic
approach based on a random walk representation (see Nguyen and P\'etr\'elis
(2013)) allowed for a sharp determination of the asymptotics of the free energy
close to criticality (see Carmona, Nguyen and P\'etr\'elis (2016)). Some
scaling 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 proven
rigorously, 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 at
criticality in Carmona and P\'etr\'elis (2017a). The second part of the paper
starts with the description of four open questions related to physically
relevant extensions of IPDSAW. Among such extensions is the Interacting Prudent
Self-Avoiding Walk (IPSAW) whose configurations are those of the 2-dimensional
prudent 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 is
proven to exist rigorously.
