The aim of this project is to study random movements of different particles and rules of the movements of their big groups using elements of probability theory. To show on concrete examples that however different the mechanism of travel of the particles in gases, liquids and solid bodies is, these travels are random. With this aim, firstly the mean of random events and their probabilities, which are the basic method of research; means of Browns movement and diffusions, in different substances and bodies about random travel on line and planes are different. All of them will be used while studying random travels of particles on a plane and in space. The problem of random movements plays a great role in the probability theory and its purposes. From all the kinds of movement one dimensional and some other movements are studied more closely. For example, movement on an endless grating made up from squares. However the study of random movements on a plane or in space has some difficulties. The theory of random movements is an important and very useful branch with of probability theory. It is successfully used in studying different development processes (physics, chemistry, etc). On the probability theory the kinematic theory of gases, the diffusion theory of matters dissolved in liquid and weighed particles are based. It can be concluded that, the example of such movement can be the movement of a small particle of a solid weighed in a liquid a so-called Brown’s movement. The appearance of Brown’s movement is the cause of diffusion, i.e. percolation of one matter into another.