Since the sets are $A=\{-1,2,2m-1\}$ and $B=\{2,m^{2}\}$, and $B⊆A$,

This implies that $m^{2}=2m-1$ or $m^{2}=-1$ (ignoring the latter as it is not a real solution),

By solving the equation $m^{2}=2m-1$, we get $m=1$,

When $m=1$, the sets become $A=\{-1,1,2\}$ and $B=\{1,2\}$, which satisfies the condition,

Hence, $m=1$,

Therefore, the answer is $\boxed{1}$.

This problem requires understanding the inclusion relationship between sets to determine the value of $m^{2}$ in set $B$, and subsequently finding the value of $m$. The problem tests knowledge of set inclusion relationships and their application, as well as the definition and range of functions, with a medium level of difficulty.