8. A mapping $f$ from set $A$ to set $B$ is called a surjection if for every element $y$ in set $B$, there is at least one $x \in A$ such that $f(x)=y$. Given a five-element set $A=\{1,2,3,4,5\}$ and a three-element set $B=$ $\{\alpha, \beta, \gamma\}$. Then the number of surjections from set $A$ to set $B$ is.
8. 150 .

The total number of mappings from a five-element set to a three-element set is $3^{5}$. Among these, the mappings that are not surjective can be divided into two categories for calculation: one category is where only one element in $B$ is the image of all elements in $A$, and there are $\mathrm{C}_{3}^{1}=3$ such mappings. The other category is where only two elements in $B$ are the images of all elements in $A$, and there are $C_{3}^{2}\left(C_{5}^{1}+C_{5}^{2}+\right.$ $\left.C_{5}^{3}+C_{5}^{4}\right)=90$ such mappings. The total of the two categories is 93. Therefore, the number of surjective mappings from $A$ to $B$ is $3^{5}-93=150$.