  Let $t_{i,j}$ be the coefficient of $x^iy^j$ in the Tutte polynomial
$T(G;x,y)$ of a connected bridgeless and loopless graph $G$ with order $n$ and
size $m$. It is trivial that $t_{0,m-n+1}=1$ and $t_{n-1,0}=1$. In this paper,
we obtain expressions of another eight extreme coefficients $t_{i,j}$'s with
$(i,j)=(0,m-n)$,$(0,m-n-1)$,$(n-2,0)$,$(n-3,0)$,$(1,m-n)$,$(1,m-n-1)$,$(n-2,1)$
and $(n-3,1)$ in terms of small substructures of $G$. Among them, the former
four can be obtained by using coefficients of the highest, second highest and
third highest terms of chromatic or flow polynomials, and vice versa. We also
discuss their duality property and their specializations to extreme
coefficients of the Jones polynomial.
