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Machine learning (ML) empowers biomedical systems with the capability to optimize their performance through modeling of the available data extremely well, without using strong assumptions about the modeled system. Especially in nano-scale biosystems, where the generated data sets are too vast and complex to mentally parse without computational assist, ML is instrumental in analyzing and extracting new insights, accelerating material and structure discoveries, and designing experience as well as supporting nano-scale communications and networks. However, despite these efforts, the use of ML in nano-scale biomedical engineering remains still under-explored in certain areas and research challenges are still open in fields such as structure and material design and simulations, communications and signal processing, and bio-medicine applications. In this article, we review the existing research regarding the use of ML in nano-scale biomedical engineering. In more detail, we first identify and discuss the main challenges that can be formulated as ML problems. These challenges are classified into the three aforementioned main categories. Next, we discuss the state of the art ML methodologies that are used to countermeasure the aforementioned challenges. For each of the presented methodologies, special emphasis is given to its principles, applications, and limitations. Finally, we conclude the article with insightful discussions, that reveal research gaps and highlight possible future research directions.