Selected by in vitro techniques (SELEX, cell-SELEX), aptamers are strands of DNA or RNA molecules able to bind a wide range of targets, from small molecules to live cells, and even tissues, with high affinity and specificity. Due to their efficient targeting ability, aptamers are extensively used in different fields of applications. For example, they ensure high performance as cancer-related markers or in recognizing cancer cells. Actually, they represent a promising way for early diagnosis (biosensors) and to deliver imaging agents and drugs, in both cancer imaging and therapy (therapeutic aptamers). Aptamer-based biosensors (aptasensors) have attracted particular attention over the last decades, so as the possibility of using aptamers in disease therapy in substitution of monoclonal antibodies. The paper briefly reviews the most recent literature on this topic, both concerning the advances in biomedical applications and in the development of electrical aptasensors. The investigation concerning the bioelectronics features of aptamers, to be implemented in the development of electrical nanobiosensors, is also reviewed. To this aim, some recent results of a theoretical/computational framework for modelling the electrical properties of biomolecules (Proteotronics) are reported.