In this talk we will present recent developments on Toeplitz matrix computations. Firstly, we will introduce fast and effective numerical algorithms for computing the inverse of a triangular Toeplitz matrix. A theoretical accuracy and error analysis will be also considered. Numerical examples will be given to illustrate the effectiveness of our methods. Secondly, a fast algorithm for solving systems of linear equations with banded Toeplitz matrices will be presented. Stability of the algorithm will be discussed and its performance will be showed by numerical experiments. Finally, a fast algorithm of two-level banded Toeplitz systems of linear equations with application to image restoration will be included.