We study graded Lie algebras of maximal class over a field
F
\mathbf {F}
of positive characteristic
p
p
. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct
|
F
|
ℵ
0
| \mathbf {F}|^{\aleph _{0}}
pairwise non-isomorphic such algebras, and
max
{
|
F
|
,
ℵ
0
}
\max \{| \mathbf {F}|, \aleph _{0} \}
soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.