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The quantum Otto cycle serves as a bridge between the macroscopic world of heat engines and the quantum regime of thermal devices composed from a single element. We compile recent studies of the quantum Otto cycle with a harmonic oscillator as a working medium. This model has the advantage that it is analytically trackable. In addition, an experimental realization has been achieved, employing a single ion in a harmonic trap. The review is embedded in the field of quantum thermodynamics and quantum open systems. The basic principles of the theory are explained by a specific example illuminating the basic definitions of work and heat. The relation between quantum observables and the state of the system is emphasized. The dynamical description of the cycle is based on a completely positive map formulated as a propagator for each stroke of the engine. Explicit solutions for these propagators are described on a vector space of quantum thermodynamical observables. These solutions which employ different assumptions and techniques are compared. The tradeoff between power and efficiency is the focal point of finite-time-thermodynamics. The dynamical model enables the study of finite time cycles limiting time on the adiabatic and the thermalization times. Explicit finite time solutions are found which are frictionless (meaning that no coherence is generated), and are also known as shortcuts to adiabaticity.The transition from frictionless to sudden adiabats is characterized by a non-hermitian degeneracy in the propagator. In addition, the influence of noise on the control is illustrated. These results are used to close the cycles either as engines or as refrigerators. The properties of the limit cycle are described. Methods to optimize the power by controlling the thermalization time are also introduced. At high temperatures, the Novikov–Curzon–Ahlborn efficiency at maximum power is obtained. The sudden limit of the engine which allows finite power at zero cycle time is shown. The refrigerator cycle is described within the frictionless limit, with emphasis on the cooling rate when the cold bath temperature approaches zero.