Starting from certain rational solutions of the quantum Yang-Baxter equation, the Yangian of simple Lie algebra $\mathfrak{g}$ can be rebuilt as a quotient of a Hopf algebra subject to a single defining relation called the $RTT$-relation. Replacing this defining relation by a quaternary relation called the reflection equation leads instead to the theory of twisted Yangians. From the point of view of deformation theory, a twisted Yangian is a filtered coideal deformation of the enveloping algebra for a twisted polynomial current algebra $\mathfrak{g}[z]^\sigma$. Here $\sigma$ is an involution of $\mathfrak{g}[z]$ coming from an involution of $\mathfrak{g}$, and $\mathfrak{g}[z]^\sigma$ is the Lie subalgebra of elements fixed by $\sigma$.

In this talk I will overview the theory of twisted Yangians associated to symplectic and orthogonal Lie algebras. After defining them and recalling their main properties, I will explain what progress has been made towards obtaining a classification of their finite-dimensional irreducible modules in terms of Drinfeld polynomials.