  The paper is devoted to the study of local derivations and automorphisms of
nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with
indices of nilpotency $3$ and $4$ admit local derivation (local automorphisms)
which is not a derivation (automorphisms). Further, it is presented a
sufficient condition under which a nilpotent Lie algebra admits a local
derivation which is not a derivation. With the same condition, it is proved the
existence of pure local automorphism on a nilpotent Lie algebra. Finally, we
present an $n$-dimensional non-associative algebra for which the space of local
derivations coincides with the space of derivations.
