We consider a nearest-neighbor random walk on Z whose probability ω(x, n) to jump to the right from site x depends not only on x but also on the number of prior visits n to x. The collection (ω(x, n)) is sometimes called the “cookie environment” due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the probability to jump to the right according to the “flavour” of the cookie eaten. Assume that the cookie stacks are i.i.d. and that only the first M cookies in each stack may have “flavour”. All other cookies are assumed to be “plain”, i.e. after their consumption the walker makes unbiased steps to one of its neighbours. The “flavours” of the first M cookies within the stack can be different and dependent. We discuss recurrence/transience, ballisticity, and limit theorems for such walks. The talk is based on joint works with Dolgopyat (University of Maryland), Mountford, Zerner.