The bracket formulation of the dynamic equations in nonequilibrium thermodynamics is examined here. After a short review of the early historical development of the subject, we present an introduction to the theoretical foundations of the one-generator bracket formalism, where the one generator is the Hamiltonian. The Hamiltonian represents the system's total (extended) internal energy but, through a Legendre transform similar to that used in equilibrium thermodynamics, its derivatives can also be calculated from the available expression for the system's extended free energy. First, the conservative component of the bracket is recognized as the Poisson bracket of Hamiltonian mechanics. Its properties are briefly reviewed and justified based on its connection to Hamiltonian mechanics. The Poisson structure of the conservative bracket is also manifested in a variety of other formalisms of the dynamic equations of nonequilibrium thermodynamics, thus providing a unilying connection. Next, the nonconservative, dissipative component is presented in the one-generator formalism, a more extensive treatment of which can be found in the research monograph by Beris and Edwards [A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, Oxford University Press, New York, 1994]. We further address here some of the more recent developments that have taken place since the publication of the above-mentioned work. Hence, the two-generator form of the bracket equations that corresponds to the GENERIC framework of description is considered next. The two generators are the Hamiltonian (or total energy), which drives the conservative part of the dynamics, and the entropy, which drives the dissipative part. Differences and similarities between the two-and the one-generator formalisms are pointed out. Finally, the advantage of the use of the bracket formulation is illustrated, as a number of recent applications are reviewed.