  In this paper, we introduce the concept of the Tutte polynomials of genus $g$
and discuss some of its properties. We note that the Tutte polynomials of genus
one are well-known Tutte polynomials. The Tutte polynomials are matroid
invariants, and we claim that the Tutte polynomials of genus $g$ are also
matroid invariants. The main result of this paper and the forthcoming paper
declares that the Tutte polynomials of genus $g$ are complete matroid
invariants.
