Atomic polarizability plays an essential role in topics ranging from van der Waals interactions, state lifetimes, and indices of refraction, to next generation atomic clocks and atomic parity non-conservation experiments. Polarizability measurements, such as the ones described in this thesis, provide valuable input to these subjects and serve as benchmark tests for sophisticated atomic structure calculations. We measured the static polarizability of potassium and rubidium with record precision and 0.5% uncertainty using a Mach-Zehnder atom interferometer with an electric-field gradient. To support future precision measurements of polarizability, we developed a new atom beam velocity measurement technique called phase choppers. Using phase choppers, we demonstrated measurements of mean atom beam velocity with an uncertainty of 0.1%. We also developed a new way to probe atomic structure: a measurement of a zero-crossing of the dynamic polarizability of potassium, known as a magic-zero wavelength. We measured the first magic-zero wavelength of potassium with 1.5 pm uncertainty and established a new benchmark measurement for the ratio of the D1 and D2 line strengths. Finally, we propose the use of a resonant photoionization detector for measurements of strontium polarizability, and the use of contrast interferometry for measurements of alkali dimer tensor polarizabilities.