We study the impact of communication or reaction delays on the long-time behavior of alignment/consensus models of Cucker-Smale type. For a model where agents interact with each other through normalized communication weights, we provide sufficient conditions for asymptotic flocking, i.e., convergence to a common velocity vector. The proof is based on a construction of a suitable Lyapunov functional. For a model without normalization, we present new stability estimates for the particle flow, relating suitable delayed and non-delayed quantities. We also briefly explain how multiplicative noise affects the long-time dynamics and present results of systematic numerical simulations.