Methods are developed for calculating characteristic functions and mea
n up-crossing rates of the response of linear systems to alpha-stable
input processes. The methods are based on the integral and series repr
esentations of these processes. The integral representation resembles
the spectral representation of wide-sense stationary processes. The al
pha-stable processes are non-Gaussian and have no moments of order two
and higher. Therefore, classical methods of linear random vibration f
or calculating the mean and covariance functions of the response from
the corresponding functions of the input do not apply. Applications ar
e presented to illustrate the proposed methods of analysis.