We investigate the tropical analogues of totally positive (TP) and totally nonnegative (TN) matrices. These arise when considering the images by the nonarchimedean valuation of the corresponding classes of matrices over the field of real Puiseux series. We show that these images are characterized in terms of their $2\times 2$ minors, and can be factored into products of tropical elementary matrices. We examine the eigenvalues of a tropical totally positive matrix, obtained precisely by the images of the positive eigenvalues of its TP-lift. If time allows, we show how tropical TN-matrices associate to planar networks and TN-Grassmannian.