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The PDE backstepping approach is a potentially powerful tool for advancing design techniques for systems with input and output delays. Three key ideas are presented in this article. The first idea is the construction of backstepping transformations that facilitate treatment of delays and PDE dynamics at the input, as well as in more complex plant structures such as systems in the lower triangular form, with delays or PDE dynamics affecting the integrators. The second idea is the construction of Lyapunov functionals and explicit stability estimates, with the help of direct and inverse backstepping transformations. The third idea is the connection between delay systems and an array of other classes of PDE systems, which can be approached in a similar manner, each introducing a different type of challenge. All the results presented here are constructive. The constants and bounds that are not stated explicitly, such as the robustness margin 8 in Theorem 3 and the convergence speeds in all the theorems, are estimated explicitly in the proofs of these theorems. However, these estimates are conservative and have limited value as guidance on the achievable convergence speeds or robustness margins.