This thesis develops new adaptive filtering algorithms suitable for communications applications with the aim of reducing the computational complexity of the implementation. Low computational complexity of the adaptive filtering algorithm can, for example, reduce the required power consumption of the implementation. A low power consumption is important in wireless applications, particularly at the mobile terminal side, where the physical size of the mobile terminal and long battery life are crucial. We focus on the implementation of two types of adaptive filters: linearly-constrained minimum-variance (LCMV) adaptive filters and conventional training-based adaptive filters.For LCMV adaptive filters, normalized data-reusing algorithms are proposed which can trade off convergence speed and computational complexity by varying the number of datareuses in the coefficient update. Furthermore, we propose a transformation of the input signal to the LCMV adaptive filter, which properly reduces the dimension of the coefficient update. It is shown that transforming the input signal using successive Householder transformations renders a particularly efficient implementation. The approach allows any unconstrained adaptation algorithm to be applied to linearly constrained problems.In addition, a family of algorithms is proposed using the framework of set-membership filtering (SMF). These algorithms combine a bounded error specification on the adaptive filter with the concept of data-reusing. The resulting algorithms have low average computational complexity because coefficient update is not performed at each iteration. In addition, the adaptation algorithm can be adjusted to achieve a desired computational complexity by allowing a variable number of data-reuses for the filter update.Finally, we propose a framework combining sparse update in time with sparse update of filter coefficients. This type of partial-update (PU) adaptive filters are suitable for applications where the required order of the adaptive filter is conflicting with tight constraints for the processing power.