The centromere is the chromosomal locus required for equal and proper segregation of the sister chromatids during cell division. In humans, the centromere is assembled on highly repetitive DNA sequences, known as alpha-satellite DNA, but these sequences are thought to be neither necessary nor sufficient for specifying centromere identity. Rather, the centromere is epigenetically specified by the histone H3 variant, CENP-A, which assembles into nucleosomes at the centromere. In this thesis, we focused on defining the genetic and epigenetic requirements for new centromere establishment. We first investigated the epigenetic component of the centromere---CENP-A. To identify the molecular determinants of CENP-A required for new centromere establishment, we utilized a Lac Operator/Lac Repressor (LacO/LacI) chromosome-tethering system to artificially target CENP-A (or mutants) to an ectopic chromosomal locus. We identified three regions within CENP-A required to initiate early steps in centromere establishment: the N-terminal tail, CENP-A Targeting Domain (CATD), and the C-terminal tail. Next, we investigated whether two post-translational modifications of CENP-A (Ser68 phosphorylation and Lys124 ubiquitination) are essential for centromere establishment or maintenance. Using a novel gene-editing approach to simultaneously and differentially replace the endogenous CENP-A gene with tagged versions, we found that neither Ser68 phosphorylation nor Lys124 ubiquitination are required for CENP-A deposition or centromere maintenance. Additionally, neither of these post-translational modifications are required for early steps in new centromere formation. Finally, we investigated the genetic component of the centromere---the DNA sequence that underlies functional centromeres. By engineering a set of constructs containing an array of LacO repeats and either alpha-satellite sequences or non-alpha-satellite sequences, we found that a centromere could form on a human artificial chromosome (HAC) in two different ways. First, we found that seeding CENP-A nucleosomes onto alpha-satellite DNA could stimulate centromere formation and form a HAC. Additionally, we identified a non-alpha-satellite genomic sequence that can form a centromere on its own, leading to the first non-repetitive, complex DNA sequence competent for HAC formation. Taken together, these findings delineate the genetic and epigenetic contributions to new centromere establishment and pave the way for future studies in synthetic biology, including the construction of synthetic human chromosomes.