In the work of Gelfand and Retakh, a method known as the quasi-determinant was introduced to define determinants over non-commutative rings. This tool has been widely applied in the study of non-commutative symmetric functions, Plücker coordinates, and solutions to non-commutative integrable systems. The Pfaffian, which serves as an important algebraic tool for skew-symmetric matrices in the commutative setting, also admits a natural extension to non-commutative rings. In this talk, I will discuss the definition of quasi-Pfaffians and their applications to non-commutative orthogonal polynomials and integrable systems.