The Operator Theory is a new theory about the hierarchical organisation of complexity in nature. The theory is based on the idea that in the space of all possible processes, a small subset exists of highly specific processes through which small objects can integrate to form new, more complex objects. The Operator Theory focuses on this small subset of objects. The processes that the Operator Theory focuses on are referred to as uniform closure of the structural and functional kind. The combination of such closures is called a dual closure. Based on dual closures, and in a step by step way, the Operator Theory identifies a branching hierarchy of kinds of objects that have increasingly complex organisation. Any object of a kind that is included in this hierarchy is called an operator, and the branching hierarchy is called the Operator Hierarchy. Interestingly, there are strong indications that, in analogy with the primary and secondary structure of amino acids, the Operator Hierarchy has a secondary structure. The Operator Theory hypothesises that this secondary structure offers a means to one day predict the structure of future kinds of operators. By offering a stringent classification of the operators of different kinds, from quarks to multicellular animals, the Operator Theory can be used to contribute to discussions about fundamental concepts in science, e.g. individuality, organismality, hierarchy, life and (the prediction of) evolution.