8. Given the set $M=\{1,2,3, \cdots, n\}, n \in \mathbf{N}_{+},$ the sum of the sums of the elements of all subsets of $M$ is (the sum of the elements of the empty set $\varnothing$ is defined to be 0).
8. $n(n+1) \cdot 2^{n-2}$.

First, consider the number of subsets of $M$ that contain the element $k$. Then the set $M^{\prime}=\{x \mid x \in M$, but $x \neq k\}$ has $n-1$ elements, and the number of subsets of $M^{\prime}$ is $2^{n-1}$; when $k$ is added to these subsets, there are $2^{n-1}$ subsets of $M$ that include the element $k$. Therefore, the desired sum is $2^{n-1}(1+2+3+\cdots+n)=n(n+1) \cdot 2^{n-2}$.