Tetraploidy – the presence of four copies of the haploid chromosome complement – is common in cancer. There is evidence that ~40% of tumors pass through a tetraploid stage at some point during their development, and tetraploid cells injected in mice are more tumorigenic than their diploid counterparts. However, the reason for this increased tumorigenicity of tetraploid cells is not well established. Most routes by which cells may become tetraploid also confer cells with double the number of centrosomes, the small membraneless organelle that organizes the cell's microtubule cytoskeleton and mitotic spindle apparatus. Centrosome number homeostasis is crucial for health, and recent studies have shown inducing extra centrosomes in cells can induce tumor formation in mice. This has led some researchers to propose that the extra centrosomes that arise together with tetraploidy may be the reason that tetraploid cells are more tumorigenic. However, several anecdotal reports have found that tetraploid clones generated and grown in vitro appear to lose their extra centrosomes. Here, I investigate the population dynamics of the loss of extra centrosomes in newly formed tetraploid cells generated via cytokinesis failure. I uncover the mechanism driving the process and build a mathematical model that captures the experimentally observed dynamics. Next, I investigate karyotypic heterogeneity in newly formed tetraploid cells and their counterparts that are grown for 12 days under standard culture conditions and find that karyotypic heterogeneity has increased after 12 days of growth after tetraploidization. The day 12 'evolved' population with increased heterogeneity formed larger colonies in soft agar than newly formed tetraploid cells or diploid parental precursors and karyotype analysis of the largest soft agar colonies revealed recurrent aneuploidies shared by a subset of colonies. Finally, I investigate the effects of different culture conditions - meant to mimic various conditions in the tumor microenvironment - on the evolution of centrosome and chromosome number in newly formed tetraploid cells and identify a small subset of conditions that altered centrosome homeostasis or the fitness of tetraploid cells. / Doctor of Philosophy / The genetic information in cancer cells is often drastically altered compared to normal cells in the body. As one important example of this, the number and structure of chromosomes - the DNA structures that hold the genetic information - is often abnormal in cancer cells. Abnormal chromosome number is closely linked with cancer development, but the details of why this leads to more cancer are not clear. One important kind of chromosome number change is when a cell undergoes incomplete cell division, and the resulting cell acquires double the number of chromosomes compared to a normal cell (known as tetraploidy). Tetraploidy occurs in close to 40% of cancers and is linked with the most aggressive cases. The abnormal cell divisions that cause tetraploidy also lead to other cellular changes. One important change is that tetraploid cells also acquire double the number of the structures that organizes cell division (centrosomes). The centrosome organizes the mitotic spindle, the major apparatus that is responsible for equally distributing chromosomes to two daughter cells during the cell division process. Extra centrosomes in cells are closely linked with cancer and can lead to additionally chromosome number changes. Researchers have believed that the extra centrosomes that are acquired with doubled chromosome number may be the major reason that tetraploid cells are linked to more aggressive cancer. However, recent studies have suggested that tetraploid cells may lose their extra centrosomes, calling into question the details of the relationship between tetraploidy and tumor formation. Here, I use human cells grown in culture to understand how extra centrosomes are lost from tetraploid cells. I find that extra centrosomes in newly formed tetraploid cells promote abnormal 'multipolar' cell divisions, in which chromosomes are segregated unevenly to three or more partitions. Such divisions are often fatal to daughter cells. In some cases, the extra centrosomes can cluster to form bipolar spindles that segregate chromosomes into two equal partitions (as is normal). When forming bipolar spindles, extra centrosomes can cluster asymmetrically (three centrosomes at one pole, one at the other) or symmetrically (two centrosomes at each pole). Tetraploid cells with a normal number of centrosomes emerge when extra centrosomes cluster asymmetrically in a bipolar spindle, yielding one tetraploid daughter cell with a normal number of centrosomes. Such cells have a fitness advantage over cells with extra centrosomes, which over time are very likely to undergo fatal multipolar divisions. Thus, cells with a normal centrosome number 'take over' the population. Next, I investigate the cancer-like properties of tetraploid cells without their extra centrosomes and find that they display increased tumor-like behavior even in the absence of extra centrosomes. Finally, I investigate whether changing the conditions in which cells are grown (in ways meant to mimic different conditions that may be experienced in the body) affects whether tetraploid cells lose their extra centrosomes. We identify a small number of conditions that do influence loss of extra centrosomes. Together, these studies illuminate important details of the relationship between tetraploidy and tumor formation. This work lays the foundation to further explore and understand the relative roles of tetraploidy, extra centrosomes, and tissue environment in cancer.