(eng): This paper considers a model of one-dimensional discrete time random walk with varying transition probabilities. These probabilities depend explicitly on the previous move of the walker and implicitly on the entire walk history, making the walk a non-Markovian stochastic process. Two basic versions of the model are introduced, some of their properties are recalled and new theoretical results derived. Then, more complex variants of models are presented. Development of walks themselves as well as the properties of connected sequences of transition probabilities are illustrated also with the aid of simulations. Applications of the model in real life situations are discussed.