Reliable wireless communications often requires accurate knowledge of the underlying multipath channel. This typically involves probing of the channel with a known training waveform and linear processing of the input probe and channel output to estimate the impulse response. Many real-world channels of practical interest tend to exhibit impulse responses characterized by a relatively small number of nonzero channel coefficients. Conventional linear channel estimation strategies, such as the least squares, are ill-suited to fully exploiting the inherent low-dimensionality of these sparse channels. In contrast, this paper proposes sparse channel estimation methods based on convex/linear programming. Quantitative error bounds for the proposed schemes are derived by adapting recent advances from the theory of compressed sensing. The bounds come within a logarithmic factor of the performance of an ideal channel estimator and reveal significant advantages of the proposed methods over the conventional channel estimation schemes.