Optical frequency standards based on forbidden transitions of trapped and laser-cooled ions have now achieved significantly higher stability and greater accuracy than primary cesium clocks. At PTB we investigate an optical clock based on the electric quadrupole transition S1/2 – D3/2 at 688 THz in the 171Yb+ ion and have shown that the frequencies realized in two independent ion traps agree to within a few parts in 1016. Results from a sequence of precise measurements of the transition frequency are now available that cover a period of seven years. Combined with data obtained at NIST on the quadrupole transition in Hg+, this allows to derive a model-independent limit for a temporal drift of the fine structure constant. We prepare to observe two more optical transitions that will provide increased sensitivity to alpha variations: The electric-octupole transition S1/2 - F7/2of Yb+ at 642 THz offers a sub-hertz frequency resolution. The ratio of the 688 THz and 642 THz frequencies in Yb+ can be measured as a dimensionless number with a femtosecond laser frequency comb. Repeated measurements of this quantity permit to search for temporal variations of alpha with a sensitivity factor ≈7, the highest in any of the available combinations of optical frequency standards. Much higher sensitivity (of order 104) may be obtained in the study of the 7.6 eV nuclear transition between the two lowest states of Th-229. We have developed a concept for a highly accurate nuclear clock based on this transition and describe first steps towards the experimental realization. This work is supported by DFG, FQXi and QUEST.