Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree
d
≤
6
d\leq 6
must be regularly ramified; (2) for any integer
d
>
6
d>6
, there exists a totally ramified rational map of degree
d
d
which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree
10
10
. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree
7
7
or
8
8
, up to pre- and post-composition by Möbius transformations.