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In this paper, we propose a new framework for the design of sparse finite impulse response (FIR) equalizers. We start by formulating greedy and convex-optimization-based solutions for sparse FIR linear equalizer tap vectors given a maximum allowable loss in the decision-point signal-to-noise ratio. Then, we extend our formulation to decision feedback equalizers and multiple-antenna systems. This is followed by further generalization to the channel shortening setup which is important for communication systems operating over broadband channels with long channel impulse responses. We propose a novel approach to design a sparse target impulse response. Finally, as an application of current practical interest, we consider self far-end crosstalk cancellation on vectored very high-speed digital subscriber line systems for cellular backhaul networks.