  Theories unifying gravity and other interactions suggest the possibility of
spatial and temporal variation of physical ``constants'' in the Universe.
Detection of high-redshift absorption systems intersecting the sight lines
towards distant quasars provide a powerful tool for measuring these variations.
We have previously demonstrated that high sensitivity to the variation of the
fine structure constant $\alpha$ can be obtained by comparing spectra of heavy
and light atoms (or molecules). Here we describe new calculations for a range
of atoms and ions, most of which are commonly detected in quasar spectra: Fe
II, Mg II, Mg I, C II, C IV, N~V, O I, Al III, Si II, Si IV, Ca I, Ca II, Cr
II, Mn II, Zn II, Ge II (see the results in Table 3). The combination of Fe II
and Mg II, for which accurate laboratory frequencies exist, have already been
used to constrain $\alpha$ variations. To use other atoms and ions, accurate
laboratory values of frequencies of the strong E1-transitions from the ground
states are required. We wish to draw the attention of atomic experimentalists
to this important problem.
  We also discuss a mechanism which can lead to a greatly enhanced sensitivity
for placing constraints on variation on fundamental constants. Calculations
have been performed for Hg II, Yb II, Ca I and Sr II where there are optical
transitions with the very small natural widths, and for hyperfine transition in
Cs I and Hg II.
