The Sylvester equation, $AX+XD=E$, is one of the most popular matrix equations in mathematics, and it arises in many applications. It can be seen as a particular case of the generalized Sylvester equation, $AXB+CXD=E$, and it is also closely related to the $\star$-Sylvester equation, $AX+X^\star D=E$, with $\star$ being the transpose or the conjugate transpose. This last equation is in turn a particular case of the generalized $\star$-Sylvester equation, $AXB+CX^\star D=E$. All these equations, that we gather under the general name Sylvester-like equations, have been considered in the literature by different authors. In this talk, we will review several results related with the solution of these equations, as well as with systems of Sylvester-like equations. In particular, we will talk about:

• Necessary and sufficient conditions for consistency.

• The dimension of the solution space.

• Necessary and sufficient conditions for uniqueness of solution.

We will also review several applications of some Sylvester-like equations to issues like the theory of orbits or block (anti)-diagonalization.