The economy as a whole and most of its constituent parts, like markets, government institutions, firms, or households, are inherently complex conceptual constructions. Micro-level diversity, decentralized interaction, self-organization, adaptation and learning, emergence, and evolution, are some of the fundamental features that the above entities share and that allow to classify them as being complex entities. In a complex economic system, existing structures of interaction are in constant mutation as individual agents contact and influence one another and, by doing so, reshape the macro environment in which socio-economic relations unfold. Notwithstanding the observed pervasiveness of complexity in economics, there are a few areas of economic thought where the discussion on the theme has gained an exceptional relevance. In this article, six of such areas are identified and their complex nature is highlighted and scrutinized. These pertain to: (i) Knowledge interactions and technological innovation; (ii) Corporate design and organizational learning; (iii) Public policies directed at market regulation; (iv) Banking and financial markets; (v) Environmental economics, sustainability, and climate change; and (vi) income inequality.