The article gives an overview of the history of formal argumentation, focusing on two main aspects: argumentation-based inference and argumentation-based dialogue. Argumentation-based inference systems help draw conclusions from incomplete or uncertain information, defining a type of logical consequence. On the other hand, argumentation-based dialogue models verbal interactions to resolve conflicts of opinion, with defined rules and strategies. The article reviews the main formal and computational models for both aspects of argumentation and discusses some key application areas.