Extreme events (i.e. terrorist attacks, vehicle impacts, explosions, etc.) often cause local damage to building structures and pose a serious threat when one or more vertical load-bearing components fail, leading to the progressive collapse of the entire structure or a large part of it. Since the beginning of the 21st century there has been growing interest in the risks associated with extreme events, especially after the attacks on the Alfred P. Murrah Federal Building in Oklahoma in 1995 and on the World Trade Center in New York in 2001. The accent is now on achieving resilient buildings that can remain operational after such an event, especially when they form part of critical infrastructures, are occupied by a large number of people, or are open to the public. This paper presents an ambitious review that describes all the main advances that have taken place since the beginning of the 21st century in the field of progressive collapse and robustness of buildings. Widely diverse aspects are dealt with, including: (1) a collection of conceptual definitions, (2) bibliometric details, (3) the present situation and evolution of codes and design recommendations, (4) quantification of robustness, (5) assessing the risk of progressive collapse, (6) experimental tests, (7) numerical modelling, and (8) research needs. Considering the comprehensive range of these aspects, this paper could be of great use to professionals and researchers who intend to enter the field of the progressive collapse of building structures and also to other experts who require an extensive and up-to-date view of this topic.