The new generation of giant ground-based telescopes will see their first light this decade. These state-of-the-art facilities will significantly surpass the resolving power of modern space-based observatories such as the James Webb telescope, thanks to their enormous aperture size and adaptive optics (AO) facilities. Without AO, atmospheric turbulence would degrade the image quality of these enormous telescopes to that of a 50 cm amateur one. These extremely large telescopes (ELTs) will further benefit from a particular branch of AO called multi-conjugate adaptive optics (MCAO), which provides an extremely high resolving power over a much wider field of view as compared to classical AO systems. The design and fabrication of such systems, as well as their optimal use for science operation, pose a great challenge as they are an order of magnitude more complicated than current AO systems. To face such a challenge, the combined knowledge of MCAO system design and fabrication, working in tandem with scientific insights into new astronomy science cases, is an extremely valuable and essential pairing. This thesis is an effort to not only contribute to the design and fabrication of ELT MCAO facilities, but also provide guidance on the optimal method to utilize these giant telescopes to achieve unprecedented astrometric measurements. On the instrumentation side, in partnership with the National Research Council of Canada's - Herzberg Astronomy and Astrophysics Institute as well as W.M. Keck Observatory in Hawaii, I was involved in the design and fabrication of a cutting edge new wavefront sensor, which is the eye of an AO system. I performed opto-mechanical design and verification studies for components of the Keck infrared pyramid wavefront sensor (IR-PWFS) as well as the Keck Planet Imager and characterizer (KPIC) instrument, which have both been commissioned and are in science operation. Furthermore, I designed the alignment plan and participated in the modification and alignment operation of a few components on the Keck II adaptive optics bench on the summit of Mauna Kea. To pave the way for the design verification of future MCAO systems for ELTs, I proposed a new method for an old challenge in the path of AO system design and verification: a flexible method for precise intensity pattern injection into laboratory AO benches. AO benches are the backbone of instrument design and modeling. One of the challenges especially important for the future generation of MCAO systems for ELTs is the verification of the effect of shadowed regions on the primary mirror. During my PhD, I successfully demonstrated the feasibility of a new proposed method to accurately model the telescope pupil. This work was done in partnership with the Laboratoire d'Astrophysique de Marseille (LAM) in France. The method I developed at LAM will be implemented in the AO Lab at NRC Herzberg Astronomy and Astrophysics. As an observational astronomer, I focused on developing methods for making optimal astrometric measurements with MCAO-enabled telescopes. The expected unparalleled astrometric precision of ELTs comes with many unprecedented challenges that if left unresolved, would jeopardize the success of these facilities as they would not be able to reach their science goals. I used observations with the only available MCAO system in science operation, the Gemini MCAO system on the 8-meter Gemini South telescope in Chile, to develop and verify a pipeline specifically designed for very high-precision astrometric studies with MCAO-fed imagers. I successfully used the pipeline to provide the precise on-sky differential distortion of the Gemini South telescope and its MCAO facilities by looking deep into the core of globular cluster NGC~6723. Using this pipeline, I produced high quality proper motions with an uncertainty floor of $\sim 45$\,$\mu$as~yr$^{-1}$ as well as measured the proper motion dispersion profile of NGC~6723 from a radius of $\sim 10$ arcseconds out to $\sim 1$\,arcminute, based on $\sim 12000$ stars. I also produced a high-quality optical-near-infrared color magnitude diagram which clearly shows the extreme horizontal branch and main-sequence knee of this cluster. / Graduate