The thesis focusses on the developments and measurements of the Yb+ optical clock at NPL. During my period there, new ion traps, trap control electronics and optical cavities have been developed. Numerous measurement campaigns have been conducted, where we measured the absolute frequency of both optical clock transitions in Yb+ relative to caesium, for the first time ever directly measured the frequency ratio of these transitions, and performed numerous international comparisons against other optical clocks. The total systematic uncertainty of the clock is now expected to be at the mid-low 10-18 level, with published uncertainty budgets at the mid 10-17 level, making it a strong candidate for a future redefinition of the second. The exotic electron structure of the excited levels in Yb+ make it a promising system for tests of fundamental physics. The frequencies of the transitions used in the Yb+ clock are highly sensitive to changes in the fine structure constant, α, which is predicted to vary by many physical theories beyond the standard model. By combing our absolute frequency measurements with a history of atomic clock measurements, we can place new best limits on the present day time variation of α, and the proton-electron mass ratio, µ.