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This study reviewed existing research on financial robo-advisors, focusing on the computational mechanisms from the artificial intelligence (AI) field, the methodological and ethical aspects of the proposed computational mechanisms, and the persuasive technology used in robo-advisors to convince or induce a specific financial activity in a person. We analyzed 48 financial robo-advisors, their algorithms (mainly machine learning) and their purposes (stock selection, user preference learning, etc.). We found a concerning lack of transparency regarding the data used to train these algorithms and a disregard for established trustworthiness criteria. Additionally, the persuasive design elements employed by robo-advisors were not well-explained in the reviewed literature. Notably, the study highlights the absence of social support principles (comparison, competition) commonly used in persuasive design. This work contributes to the financial AI community by suggesting better methodological approaches when AI-based persuasion is used.