Over the past decades there have been exciting and rapid developments of highly specific molecules to bind cancer antigens that are overexpressed on the surfaces of malignant cells. Nanomedicine aims to exploit these ligands to generate nanoscale platforms for targeted cancer therapy, and to do so with negligible off-target effects. Aptamers are structured nucleic acids that bind to defined molecular targets ranging from small molecules and proteins to whole cells or viruses. They are selected through an iterative process of amplification and enrichment called SELEX (systematic evolution of ligands by exponential enrichment), in which a combinatorial oligonucleotide library is exposed to the target of interest for several repetitive rounds. Nucleic acid ligands able to bind and internalize into malignant cells have been extensively used as tools for targeted delivery of therapeutic payloads both in vitro and in vivo. However, current cell targeting aptamer platforms suffer from limitations that have slowed their translation to the clinic. This is especially true for applications in which the cargo must reach the cytosol to exert its biological activity, as only a small percentage of the endocytosed cargo is typically able to translocate into the cytosol. Innovative technologies and selection strategies are required to enhance cytoplasmic delivery. In this review, we describe current selection methods used to generate aptamers that target cancer cells, and we highlight some of the factors that affect productive endosomal escape of cargoes. We also give an overview of the most promising strategies utilized to improve and monitor endosomal escape of therapeutic cargoes. The methods we highlight exploit tools and technologies that can potentially be incorporated in the SELEX process. Innovative selection protocols may identify aptamers with extended biological functionalities that allow effective cytosolic translocation of therapeutics. This in turn may facilitate successful translation of these platforms into clinical applications.