The idea of this paper is to interest readers in sequence spaces and matrix transformations as the starting point for possible applications in operator theory. In this paper we give survey of the known results on the so--called classical sequence spaces - the sets $\ell_{\infty}$, $c$ and $c_{0}$ of bounded, convergent and null sequences. We consider their basic properties, $\beta$-duals and the characterizations of matrix transformations between them. After that, we establish some results related to general linear operators from the space $c$ into each of the classical spaces. Furthermore, we characterize the classes of compact operators between them, applying two approaches for that purpose - the Hausdorff measure of noncompactness and Sargent's results \cite{sarg}. Presented results together with all the other known results about compactness, will close some gaps in the existing literature. All these results are collected in the same place and can be a useful start for further research. We also preesnted some possible ideas for further work in the area of doubly stochastic operators.