The research presented in this thesis is on inverse problems encountered in the field of signal processing. Theory, classification and solution techniques of linear discrete inverse problems (LDIP) are investigated. LDIP are classified as underdetermined LDIP (ULDIP) and overdetermined LDIP (OLDIP). The solution methods developed for LDIP are applied to the particular problems of signal processing mainly channel estimation, equalization and compressive sampling. A new solution technique named constraint removal (CR) is presented for ULDIP type problems with sparse inputs. CR is applied to terrestrial digital TV (DTV) channel estimation. CR is also compared with subspace pursuit (SP) and linear programming. Regularization and optimum regularization parameter selection for ill-posed OLDIP type problems are discussed. Sparse channel estimation for wireless digital communications is investigated. A new channel estimation method, permuted deconvolution (PDEC), for long delay spread channels with short training sequences is proposed and compared with other methods. A review on equalization is presented. Different equalization techniques are discussed and compared. DFE is explained from an inverse problem perspective. A new non-feedback equalization technique called frequency compensated linear equalization (FC-LE) for sparse channels is presented and compared with DFE.