In this chapter, after recalling two historical examples, we introduce the notion of duality between two Markov processes. We define both semigroup duality and generator duality, followed by a discussion on their reciprocal relation. We then consider the notion of duality between two algebras of operators in a more general context, i.e. beyond Markov processes. We proceed by increasing levels of generality: first we treat algebras of matrices, then algebras of bounded operator on a Hilbert space. Finally we give the abstract formulation of duality between two algebras which arises by considering two intertwined representations. The added value of this abstract formulation is that dualities of Markov processes can be understood from algebra representation theory. As an example, we show how this works for the duality between the Wright-Fisher diffusion and the Kingman’s coalescent block counting process, which is explained using two representations of the Heisenberg Lie algebra.