We prove some "universality" results for topological dynamical systems. In particular, we show that for any continuous self-map T of a perfect Polish space, one can find a dense, T -invariant set homeomorphic to the Baire space N N ; that there exists a bounded linear operator U : ℓ 1 → ℓ 1 such that any linear operator T from a separable Banach space into itself with T ≤ 1 is a linear factor of U ; and that given any σ-compact family F of continuous self-maps of a compact metric space, there is a continuous self-map U F of N N such that each T ∈ F is a factor of U F . Throughout the paper, by a map we always mean a continuous mapping between topological spaces. For us, a dynamical system will be a pair (X, T ), where X is a topological space and T is a self-map of X.A dynamical system is (X, T ) is a factor of a dynamical system (E, S), and (E, S) is an extension of (X, T ), if there is a map π from E onto X such that T π = πS, i.e. the following diagram commutes:When T and S are linear operators acting on Banach spaces X and E, we say that T is a linear factor of S if the above diagram can be realized with a linear factoring map π.