It is now generally accepted that chromosomes in the cell nucleus are organized in distinct domains, first called chromosome territories in 1909 by the great cytologist Theodor Boveri. Yet, even today chromosomes have remained enigmatic individuals, whose structures, arrangements and functions in cycling and post-mitotic cells still need to be explored in full detail. Whereas numerous recent reviews describe present evidence for a dynamic architecture of chromosome territories and discuss the potential significance within the functional compartmentalization of the nucleus, a comprehensive historical account of this important concept of nuclear organization was lacking so far. Here, we describe the early rise of chromosome territories within the context of the discovery of chromosomes and their fundamental role in heredity, covering a period from the 1870th to the early 20th century (part I, this volume). In part II (next volume) we review the abandonment of the chromosome territory concept during the 1950th to 1980th and the compelling evidence, which led to its resurrection during the 1970th to 1980th.