In this section, the synthesis of saturated N-and O-heterocycles via formal cycloaddition is presented. The main focus is on metal-catalyzed reactions involving C-C or C-X bond cleavage in three-or four-membered rings. After a fast presentation of pioneering works, the important breakthroughs of the last two decades are presented. The section starts with reactions involving three-membered rings. Formal [3+2] cycloadditions of donor-acceptor-substituted cyclopropanes and methylenecyclopropanes with carbonyls and imines are important methods to access tetrahydrofuran and pyrrolidine heterocycles. Formal [3+3] cycloadditions have emerged more recently. On the other hand, reactions of epoxides and aziridines with carbon monoxide or cumulenes are now well-established method to access heterocycles. These processes have been completed more recently with cycloaddition with olefins, carbonyls and imines. The section ends with the emerging field of four-membered ring activation for cycloaddition with systems. This is the peer reviewed version of the following article: Top. Heterocycl. Chem. 2013, 32, 225, which The discovery of classical cycloaddition reactions, such as the (hetero) DielsAlder and 1,3-dipolar cycloadditions, has contributed tremendously to a more efficient access towards both carbocycles and heterocycles. The introduction of the term cycloaddition was necessary to distinguish these new types of reactions from previously discovered processes leading to cyclic structures, such as the famous Robinson annulation. In principle, each cycloaddition can be considered as a special case of the more general annulation process, but from which point on an annulation can be called a cycloaddition has been the topic of intensive discussions for decades, and is still not settled today. In 1968, Huisgen proposed a set of rules for the definition of cycloaddition, and the two first are still largely recognized as prerequisite: Although the rule that all the atoms of the starting materials have to be included in the product is not explicitly included in the definition, this requirement is usually recognized by most organic chemists. Even if electrocyclic cyclization processes were included in the original definition of Huisgen, the term cycloaddition is mostly used today for those reactions proceeding via the formation of at least two new bonds. Nevertheless, several researchers think that the term cycloaddition should be more strictly limited to reactions involving a continuous overlap of electrons, and consequently allowing a concerted process. In fact, in his seminal publication, Huisgen already introduced further rules, in particular rule number 3, which explicitly stated that cycloadditions should not involve the cleavage of sigma bonds: -Huisgen Rule 3: "Cycloadditions do not involve the cleavage of bonds." Unfortunately, in the same publication, Huisgen also described several reactions proceeding via -bond cleavage as cycloaddition. To solve this definition dilemma, several researchers have used th...