  Numerous error estimates have been carried out on various numerical schemes
for subdiffusion equations. Unfortunately most error bounds suffer from a
factor $1/(1-\alpha)$ or $\Gamma(1-\alpha)$, which blows up as the fractional
order $\alpha\to 1^-$, a phenomenon not consistent with regularity of the
continuous problem and numerical simulations in practice. Although efforts have
been made to avoid the factor blow-up phenomenon, a robust analysis of error
estimates still remains incomplete for numerical schemes with general
nonuniform time steps. In this paper, we will consider the $\alpha$-robust
error analysis of convolution-type schemes for subdiffusion equations with
general nonuniform time-steps, and provide explicit factors in error bounds
with dependence information on $\alpha$ and temporal mesh sizes. As
illustration, we apply our abstract framework to two widely used schemes, i.e.,
the L1 scheme and Alikhanov's scheme. Our rigorous proofs reveal that the
stability and convergence of a class of convolution-type schemes is
$\alpha$-robust, i.e., the factor will not blowup while $\alpha\to 1^-$ with
general nonuniform time steps even when rather general initial regularity
condition is considered.
