In this paper, we consider parameter estimation of univariate α-stable distributions and their mixture. We propose firstly an estimation method based on the characteristic function by the use of a Gaussian kernel estimator of the density distribution. The choice of the optimal bandwidth parameter was done using a plug-in method. We highlight another estimation procedure for the maximum likelihood framework based on the false-position algorithm to find a numerical root of the log-likelihood through the score functions. In the case of a mixture α-stable distributions, the EM algorithm and the Bayesian estimation method were modified in order to propose an efficient tool for parameter estimation. Although we have limited the mixture study to two components, the proposed methods can be generalized to several components of mixture. A simulation study is conducted to evaluate the performance of our methods which are then applied to real data. Our results seems accurately estimating mixture of α-stable distributions. The application concerned the estimation of reproduction Parameter estimation for stable distributions and their mixture number of the Covid-19 in Mayotte and the Enzyme dataset density distribution. We compare the proposed methods along with a detailed discussion, and we conclude with some other forthcoming works.