  For a linear system $|C|$ on a smooth projective surface $S$, whose general
element is a smooth, irreducible curve, the Severi variety $V_{|C|, \delta}$ is
the locally closed subscheme of $|C|$ which parametrizes irreducible curves
with only $\delta$ nodes as singularities. In this paper we give numerical
conditions on the class of divisors and upper-bounds on $\delta$ ensuring that
the corresponding Severi variety is everywhere smooth of codimension $\delta$
in $|C|$ (regular, for short). In particular, we focus on surfaces of general
type, since for such surfaces less is known than what is proven for other
cases. Our result generalizes some results of Chiantini-Sernesi (1997) and of
Greuel-Lossen-Shustin (1997 - in the case of nodes) as it is shown by some
examples of Severi varieties on blown-up surfaces or surfaces in $\P^3$ which
are elements of a component of the Noether-Lefschetz locus. We also consider
examples of regular Severi varieties on surfaces in $\P^3$ of general type
which contain a line.
