Geographical flow can be defined as the movements of geographical objects between different locations, which are usually displayed as the movement of matter, information, energy and funds, e.g. the jobs-housing flow in a city, communications between different mobile phone holders and the fund transferred between different business entities. Due to the existence of the various flows, the link strength between different locations may not depend on distance only, say one may strongly related to a store faraway through express delivery rather than a store nearby. The traditional knowledge of distance-decay law may be changed. As a result, research on the geographical flow may help to understand geographical patterns and their mechanism from a new point of view. Two conceptual models are introduced for the expression of geographical flows in this paper. In the first model, a flow is abstracted as a coordinate quaternion composed of the origin point and the destination point (called the orthonormal flow model). Thus, the flow space can be defined as a 4-D space which is formed by the Cartesian product of two 2-D spaces. In the second model, a flow is composed of the origin point coordinates, the flow length and the flow angle (called the polar coordinate model). Based on the expression models, four distances are defined, specifically, maximum distance, additive distance, average distance and weighted distance. In addition, this paper defines some other flow measurements, including flow direction, the volume of a flow's r-neighborhood and the flow density. According to the combination of different statistical features (i.e. heterogeneity, homogeneity and randomness) between variables in the polar coordinate model, the spatial patterns of geographical flows are divided into six single patterns including random, clustering, convergent and divergent, community, parallel (angle-clustered) and equilong (length-clustered). The methods for identifying different flow patterns are also analyzed and summarized. Besides the single patterns, the combination of different single patterns will generate mixed patterns, and if more than one type of flows coexists, multi-flow patterns can be produced. Regarding research directions of geographical flow in the future, three aspects should be given more attentions: the basic statistical theory of flow, the mining method of flow pattern and its application in practical problems.