In the talk, we first give the overview and classification of various approaches for defining 2D discrete surfaces in a 3D discrete space. We also analyze the difficulty of the problem caused by the spatial discreteness and discuss the advantages and disadvantages of each approach. For our discrete surface representation which provide a good solution for the border tracking problem, we apply some basic ideas of combinatorial topology and algebraic topology, which are also used in some of previous work (but in a different way). After obtaining a discrete surface representation, we consider an algorithm for tracking the border which is always guaranteed to have a surface structure. We also gives some interesting properties of our tracked border and make comparison with the results by other border tracking algorithms.