In this thesis we will consider Markov operators on cones . More precisely, we let X equipped with certain norm be a real Banach space, K in X be a closed, normal cone with nonempty interior, e in Int (K) be an order unit. A bounded, linear operator T from X into X is a Markov operator w.r.t. K and e if K is invariant under T and e is fixed by T. We consider then the adjoint of T, T* and homogeneous, discrete time Markov system given by u_k+1 = T*(u_k), k = 0,1,2 where u_0(x) is nonnegative for all x in K and u_0 (e ) = 1.The final goal of the theoretical part of this thesis is to give a sufficient on T that will guarantee the converegnce of the Markov system given above to some unique,invariant measure. This is done in theorem 6.1 which states that if T is strict contraction w.r.t. a certain norm, then this is sufficient condition for the convergence of the Markov system. The theorem states that same condition on T is also a sufficient condition for the convergence of the system x_k+1 = T(x_k) k= 0,1,2 to converge to a scalar multiple of e, so called consensus state. We apply this theorem to stochastic matricies, to Markov operators acting on the space of all continuous,real valued functions on some compact, Hausdorff topological space and to Kraus maps acting on the space of all n*n Hermitian matricies.