Malcev and independently Neumann and Taylor have shown that nilpotent groups can be defined by using semigroup identities. This leads to the notion of a nilpotent semigroup (in the sense of Malcev). In this talk finite semigroups that are close to being nilpotent will be investigated. Obviously every finite semigroup that is not nilpotent has a subsemigroup that is minimal for not being nilpotent, i.e., every proper subsemigroup and every Rees factor semigroup is nilpotent. It is called a minimal non-nilpotent semigroup.