  In this article, we first review the connection between L\'evy processes and
infinitely divisible random variables, and the classification of infinitely
divisible distributions. Using this connection and the L\'evy-Khinchine
representation of the characteristic function, we establish a Stein identity
for an infinitely divisible random variable. The classification and slight
modification in approach give us a Stein identity for an $\alpha$-stable random
variable with $\alpha\in (0,2).$ Using fine regularity estimates for the
solution to Stein equation, we derive error bounds for $\alpha$-stable
approximations. We then apply these results to obtain rates of convergence.
Finally, we compare these rates with the results available in the literature.
