We present preliminary results of two surveys of weak Mg II (rest frame equivalent width $W_r$(2796) $<$ 0.3Å) absorbers at $\langle z \rangle$=0.15 and at $\langle z \rangle$=1.75. For the low redshift survey, we used 25 HST/STIS echelle quasar spectra ($R$=45,000) that covered Si II$\lambda$1260 and C II$\lambda$1335 at 0 $<z<0.3$. These transitions were used as a tracer of Mg II$\lambda$2796, allowing a survey of redshift path-length $g(z)$=5.3. Our high redshift survey covers a redshift path-length of $g(z)$=6.7 in 15 high resolution QSO spectra obtained with UVES on VLT. We find that $dN/dz=0.96\pm 0.19$ for $0<z<0.3$ and $0.02<W_r$(2796) $<$ 0.3Å, and $dN/dz=1.05\pm 0.16$ for $1.4<z<2.2$ over the same equivalent width range. These numbers are to be compared to the results of a previous survey of weak Mg II absorbers: $dN/dz=1.74\pm 0.10$ for $0.4<z<1.4$ (Churchill et al. 1999). There appears to be a peak epoch for weak Mg II absorbers at $z\sim$0.9. At higher redshifts, the observed value is significantly less than expected for cosmological evolution, and somewhat smaller than expected considering the larger extragalactic background radiation at $\langle z\rangle$=1.75. We conclude from this that the processes that produce weak Mg II absorbing structures are equally, or somewhat less, active at $\langle z\rangle$=1.75 than at $\langle z \rangle$=0.9. At lower redshifts, $\langle z\rangle$=0.15, we would expect a larger number of weak Mg II absorbers than observed (comparing to intermediate redshifts). A significant fraction of these expected objects would result from the evolution of low density structures in which only C IV absorption was observable at higher redshifts. The overall evolution of weak Mg II absorbers is consistent with them being related to some type of global star formation activity, perhaps restricted to certain environments. More detailed examination of the evolution of the properties of weak Mg II absorbers is needed.