This was a unifying theory of genetics that in 1902 stated that chromosomes were the carriers of the genetic material, the genes of Mendel. It also stated that chromsomes were linear and that genes were distributed along them. It was proposed independently by Theodore Boveri (1862-1915) and Walter Sutton (1877-1916).
At this time many researchers were working in genetics as Mendel's data had just resurfaced and cellular biology was making headway in the study of chromosomes in both mitosis and meiosis. Mitosis is when a cell divides in two and thus separates its pairs of chromosomes. Meiosis is when germ cells form via division and the cell stays with the half number of chromosomes ready for fertilization. It was therefore possible for both Boveri and Sutton to come up with and publish correct theories without hard core proof. Truth was in the air.
Boveri had some own data at his disposal though. He showed that sea urchins had to have all chromosomes to develop embryologically. Sutton's work with grasshoppers showed that chromosomes occur in matched pairs of maternal and paternal chromosomes which separate during meiosis.
However, Thomas Hunt Morgan (1866-1945) is credited for furnishing hard core proof in 1910 in the form of genetic linkage studies in the fruit fly Drosophila melanogaster, a species that he made into the pet animal of geneticists. He was the first scientist to get the Nobel Prize in Physiology and Medicine in genetics in 1933 "for his discoveries concerning the role played by chromosomes in heridity".
Morgan received his PhD from Johns Hopkins University in 1890 and studied embryology at his tenure in Bryn Mawr. Following the rediscovery of Mendelian genetics in 1900, Morgan began studies on mutations of fruit flies. At Columbia University he was then able to prove that genes are on chromosomes. He had thus, among other things, opened the quest for the biochemical identity of the gene.
As I earlier noted Miescher had found DNA 1869, a weak acid in extracts from nuclei of white blood cells, but this finding had not caught enough attention because it was not functionally defined. Because it was not until 1944 that Oswald Avery, Colin McLeod and Maclyn McCarty isolated DNA, then called the transforming principle.
Data from Wikipedia