In this paper we studied totally positive and strictly totally positive matrices. In the first chapter we introduced basic concepts, examples, discussed some operations that preserve the considered classes of matrices as well as the methods for constructing new matrices of the same class from the existing ones. In particular, we discuss how elementary transformations over rows and columns of a matrix effect the (strict) total positivity of matrices, we prove that transposition preserves (strict) total positivity, and show how to modify the inverse of a (strictly) totally positive matrix, which does not have to be a (strictly) totally positive matrix, to a matrix with this property, etc. Since the definition of (strict) total positivitness requires verification of the positivity or nonnegativity of all minors of a given matrix, it is desirable to find sufficient conditions which reduce the number of these verifications. Some of the results of this type are Fekete’s lemma and the Gasca-Penna theorem. In the last chapter we have provided examples of known matrices belonging to the considered class of matrices. The best known example is the Pascal matrix, whose elements are binomial coefficients. Further, the Cauchy and the Vandermonde matrix, with the suitably chosen entries, are also (strictly) totally positive matrices. In the last section, we studied Jacobi, Hankel and Toeplitz matrices.