This chapter considers the design of various types of functional observers for a class of time-delay systems with multiple delays in both state and output vectors. By a novel idea of estimating directly a new generalized functional that has multiple state-delayed terms, we can relax their observer existence conditions. Through a numerical example, we show the significance of the new estimation problem, where we can expand the dimension of the designed observer beyond the original dimension of the state vector. This new finding offers an effective way to design generalized functional observers for a wide class of time-delay systems.