In recent years, motivated by the application, many mathematicians have been focusing their research on random matrices of growing dimensions. This Master’s Thesis describes the reason why great attention is paid to the sample covariance and correlation data matrices and their eigenvalues and the corresponding eigenvectors. In the first chapter, in the context of so-called Gaussian data matrices, a brief review of the results on the asymptotic behaviour of the largest eigenvalues and corresponding eigenvectors of random covariance matrices with growing dimensions is presented. Also, an overview of the results in the case of the distributions which satisfy regular variation condition was made, where the development of the theory progressed more slowly. Chapters afterwards contain recent results in the case of distributions which satisfy regular variation condition. The second chapter contains the results of the newest research on the asymptotic behavior of the largest eigenvalue of the sample covariance matrix for data matrix with independent and identically distributed components, coming from heavy tailed distribution. Also, the results on the eigenvectors associated with the largest eigenvalues, as well as the generalization on the autocovariance matrices, are presented. The third chapter introduces the dependence between the rows and columns of the data matrix. Furthermore, it is presented that asymptotic convergence of the largest eigenvalue can be obtained from the convergence of the corresponding point processes. Finally, the illustrations on simulated and empirical data of theoretical results from the preceding chapters are given in the fourth chapter. Lastly, a brief analysis of the time series components of the S&P 500 stock index was conducted.