  In this paper we study some properties of random walks perturbed at extrema,
which are generalizations of the walks considered e.g., in Davis (1999). This
process can also be viewed as a version of {\em excited random walk}, studied
recently by many authors. We obtain a few properties related to the range of
the process with infinite memory. We also prove the Strong law, Central Limit
Theorem, and the criterion for the recurrence of the perturbed walk with finite
memory.
