Poincare at the beginning of XX-th century, and then Carleman, Koopman and von Neumann in the 1920-is made their visionary contributions to the analysis of dynamical systems behavior through the analysis of the spectral properties of the associated Koopman operator. In this century the interest for the Koopman operator theory and applications is renewed thanks to the advances of the functional analysis as well as development of data-driven algorithms. Originally Koopman operators were aimed at ergodic theory of measure- preserving systems. Today applications to non-autonomous dynamical systems or dynamical systems in presence of uncertainty are of highest interest. In this work we present results on the basic properties of the eigenvalues and eigenfunctions of the non-autonomous Koopman operators as well as the analysis of issues that arise when data-driven algorithms are applied to the evaluation of the non-autonomous Koopman eigenvalues and eigenvectors. The rst data-driven approach is DMD application to moving windows of snapshots. In such approach all DMD methods manifest signi cant errors and we analyze and prove the structure of these errors. The second data- driven approach is DMD application to large Hankel matrices of snapshots. In this approach we investigate the relation between the nonautonomous Koopman operator eigenvalues and eigenfunctions and the eigenvalues and eigenfunctions of the underlying extended autonomous dynamical system. We illustrate the results of our analysis on several synthetic test-examples.