  We investigate the cosmological evolution of interstellar dust with a
semi-analytical galaxy formation model ($\nu^2$GC), focusing on the evolution
of grain size distribution. The model predicts the statistical properties of
dust mass and grain size distribution in galaxies across cosmic history. We
confirm that the model reproduces the relation between dust-to-gas ratio and
metallicity in the local Universe, and that the grain size distributions of the
Milky Way (MW)-like sample become similar to the so-called MRN distribution
that reproduces the observed MW extinction curve. Our model, however, tends to
overpredict the dust mass function at the massive end at redshift $z\lesssim
0.8$ while it reproduces the abundance of dusty galaxies at higher redshifts.
We also examine the correlation between grain size distribution and galaxy
properties (metallicity, specific star formation rate, gas fraction, and
stellar mass), and observe a clear trend of large-grain-dominated,
small-grain-dominated, and MRN-like grain size distributions from unevolved to
evolved stages. As a consequence, the extinction curve shapes are flat, steep,
and intermediate (MW-like) from the unevolved to evolved phases. At a fixed
metallicity, the grain size distribution tends to have larger fractions of
small grains at lower redshift; accordingly, the extinction curve tends to be
steeper at lower redshift. We also predict that supersolar-metallicity objects
at high redshift have flat extinction curves with weak 2175 \AA bump strength.
