Current data acquisition methods acquire massive amounts of data, which needs to be compressed, i.e. part of these data is discarded, in order to allow its storage or transmission. This is a wasteful and excessive work. The new emerging compressive sensing (CS) concept enables directly representing the signals with far fewer measurements than required by current techniques, and resolving the signals from these fewer measurements. This is made possible by exploiting the inherent redundancy of the signals, which is translated to sparsity in a transform domain. CS can recover the sparse signal accurately using various techniques: convex optimization algorithms (i.e. Basis Pursuit (BP)), greedy algorithms (i.e. Orthogonal Matching Pursuit (OMP)) and projected gradient descent algorithms (i.e. Iterative Hard Thresholding (IHT)). In classical CS the resulting measurements values are real values and need to be quantized. As the quantization part is the main burden and main source of error in the analog to digital converter (ADC) process, the quantized CS decreases the bit-depth, and reveals in its extreme case, the 1-bit compressive sensing (1-bit CS) framework, its capability to represent each measurement value using one bit only, i.e. its sign. In the 1-bit CS configuration, the quantizer can be a low power simple comparator running at high rate, producing a positive or negative sign for each measurement value. This reduces the complexity of the signal acquisition. In this thesis we present the classical CS concept and its main recovery algorithms, and then the 1-bit CS formulation is illustrated. We address mainly in this thesis the main recovery algorithms for the 1-bit CS recovery problem: the first proposed algorithms, the Renormalized Fixed Point Iteration (RFPI) and Binary Iterative Hard Thresholding (BIHT), are suitable for noise free measurements. The Binary Iterative Hard Thresholding (BIHT) and the HISTORY algorithms can reconstruct the sparse signal from noisy measurements. The probabilistic Bayesian recovery algorithm does not need the sparsity value as an input. The performance of these algorithms is analyzed under different sparsity values and under different noisy measurement scenarios, with a comparison between them.