The precision of atomic clocks continues to improve at a rapid pace: While caesium clocks now reach relative systematic uncertainties of a few 10-16, several optical clocks based on different atomic systems are now reported with uncertainties in the 10-18 range. This variety of precise clocks will allow for improved tests of fundamental physics, especially quantitative tests of relativity and searches for variations of constants. Laser-cooled and trapped ions permit the study of strongly forbidden transitions with extremely small natural linewidths and long coherence times. The frequency of the electric octupole transition S1/2 - F7/2 at 467 nm in 171Yb+ with a natural linewidth in the nHz range is remarkably insensitive against external electric and magnetic fields. We evaluate the systematic uncertainty of a frequency standard that is based on this transition as 4*10-18 at present. An even better isolation from external perturbations can be expected for the nuclear transition in 229Th3+ at about 160 nm with an expected linewidth in the mHz range. In order to excite the so far only indirectly observed nuclear transition using electronic bridge processes, we investigate the dense electronic level structure of Th+. Both transitions, in Yb+ and 229Th, are predicted to be highly sensitive to changes in the fine structure constant. I will give an update on limits on variations of constants as obtained from atomic clock comparisons.