For almost a century the famous theorem of Shannon-Nyquist has been very important in digital signal processing applications as the basis for the number of samples required to efficiently reconstruct any type of signal, such as speech and image data. However, signals and images are mainly stored and processed in huge files, which require more storage space, they take longer to transmit and they demand a large computational cost for processing. For this purpose many compression techniques have been introduced including the emerging field of Compressed Sensing (CS). CS is a novel and fast sampling and recovery process, which has attracted considerable research interest with several new application areas. By exploiting the signal and the measurements structure we are able to recover a signal from what was previously considered as highly under-sampled measurements, according to the Shannon- Nyquist criterion. This reconstruction is accomplished by finding the sparsest solution for an ill-posed system of linear equations, which is an NP-hard combinatorial optimisation problem. This thesis focuses on the l0-norm based minimisation problem which arises from sparse signal or image recovery, using the CS technique. A new, fast heuristic is proposed to directly minimise a continuous function of the l0 norm. This swarm based stochastic method provides better sparse solutions for highly under-sampled and over-sampled cases even under the presence of noise with small error and less time complexity, compared with several well-known competing approaches. The evaluation methodology includes different parameters of the l0-heuristic and is based on measuring recovery error and execution time under various sparsity levels, sample sizes, sampling matrices and transform domains. The mathematical background of CS, including the key aspects of sparsity and incoherence in measurements are also provided, together with applications and further open research questions, such as weakly sparse signals in noisy environments.