Two-sided matrix equation $AX = XB$

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How do you find a square matrix $X$ from an equation of the type

$AX = XB$? ($A$ and $B$ are known square matrices)

How do I scale the solution up to multiple $A$s and $B$s for a single unknown $X$, i.e.

$\forall_{i=1...n} A_i X=XB_i$

Besides finding a solution, I would like to know what the problem is called. I am sure it pops up in lots of different research areas, and must have been solved before, but I don't know what exactly to ask Uncle Google.

Thanks you all in advance.