The theoretical problem of finding the solution to an underdeterminedset of linear equations has for several years attracted considerable attentionin the literature. This problem has many practical applications.One example of such an application is compressed sensing (cs), whichhas the potential to revolutionize how we acquire and process signals. Ina general cs setup, few measurement coefficients are available and thetask is to reconstruct a larger, sparse signal.In this thesis we focus on algorithm design and selected applicationsfor cs. The contributions of the thesis appear in the following order:(1) We study an application where cs can be used to relax the necessityof fast sampling for power spectral density estimation problems. Inthis application we show by experimental evaluation that we can gainan order of magnitude in reduced sampling frequency. (2) In order toimprove cs recovery performance, we extend simple well-known recoveryalgorithms by introducing a look-ahead concept. From simulations it isobserved that the additional complexity results in significant improvementsin recovery performance. (3) For sensor networks, we extend thecurrent framework of cs by introducing a new general network modelwhich is suitable for modeling several cs sensor nodes with correlatedmeasurements. Using this signal model we then develop several centralizedand distributed cs recovery algorithms. We find that both thecentralized and distributed algorithms achieve a significant gain in recoveryperformance compared to the standard, disconnected, algorithms.For the distributed case, we also see that as the network connectivity increases,the performance rapidly converges to the performance of thecentralized solution.