The monograph is devoted to modern problems of geometry. Issues close to the classical Helly theorem and related to the properties of a family of translates (i.e. images under parallel transfers) of convex bodies are considered. The authors give a complete overview (with detailed evidence) of the results in this area of ​ ​ combinatorial geometry, ranging from the classical work of the famous Hungarian mathematician B. Szekefalvi-Nagy (1954) to the most recent works.