In this paper we discuss several methodological issues related to the identification and estimation of Average Treatment on the Treated (ATT) effects in the presence of low compliance. We consider non-experimental data consisting of a treatment group, where a program is implemented, and of a control group that is non-randomly drawn, where the program is not offered. Estimating the ATT involves tackling both the non-random assignment of the program and the non-random participation among treated individuals. We argue against standard matching approaches to deal with the latter issue because they are based on the assumption that we observe all variables that determine both participation and outcome. Instead, we propose an IV-type estimator which exploits the fact that the ATT can be expressed as the Average Intent to Treat divided by the participation share, in the absence of spillover effects. We propose a semi-parametric estimator that couples the flexibility of matching estimators with a standard Instrumental Variable approach. We discuss the different assumptions necessary for the identification of the ATT with each of the two approaches, and we provide an empirical application by estimating the effect of the Mexican conditional cash transfer program, Oportunidades, on food consumption.