Modern communication and space systems such as satellite communication devices, radars, SAR and radio astronomy interferometers are realized with large antenna arrays since this kind of radiating systems are able to generate radiation patterns with high directivity and resolution. In such a framework conventional arrays with uniform inter-element spacing could be not satisfactory in terms of costs and dimensions. An interesting alternative is to reduce the array elements obtaining the so called "thinned arrays". Large isophoric thinned arrays have been exploited because of their advantages in terms of weight, consumption, hardware complexity, and costs over their filled counterparts. Unfortunately, thinning large arrays reduces the control of the peak sidelobe level (PSL) and does not give automatically optimal spatial frequency coverage for correlators. First of all the state of the art methodologies used to overcome such limitations, e.g., random and algorithmic approaches, dynamic programming and stochastic optimization algorithms such as genetic algorithms, simulated annealing or particle swarm optimizers, are analyzed and described in the introduction. Successively, innovative guidelines for the synthesis of large radiating systems are proposed, and discussed in order to point out advantages and limitations. In particular, the following specific issues are addressed in this work: 1. A new class of analytical rectangular thinned arrays with low peak sidelobe level (PSL). The proposed synthesis technique exploits binary sequences derived from McFarland difference sets to design thinned layouts on a lattice of P(P+2) positions for any prime P. The pattern features of the arising massively-thinned arrangements characterized by only P(P+1) active elements are discussed and the results of an extensive numerical analysis are presented to assess advantages and limitations of the McFarland-based arrays. 2. A set of techniques is presented that is based on the exploitation of low correlation Almost Difference Sets (ADSs) sequences to design correlator arrays for radioastronomy applications. In particular three approaches are discussed with different objectives and performances. ADS-based analytical designs, GA-optimized arrangements, and PSO optimized arrays are presented and applied to the synthesis of open-ended "Y" and "Cross" array configurations to maximize the coverage u-v or to minimize the peak sidelobe level (PSL). Representative numerical results are illustrated to point out the features and performances of the proposed approaches, and to assess their effectiveness in comparison with state-of-the-art design methodologies, as well. The presented analysis indicates that the proposed approaches overcome existing PSO-based correlator arrays in terms of PSL control (e.g., >1.0dB reduction) and tracking u-v coverage (e.g., up to 2\% enhancement), also improving the speed of convergence of the synthesis process. 3. A genetic algorithm (GA)-enhanced almost difference set (ADS)-based methodology to design thinned planar arrays with low-peak sidelobe levels (PSLs). The method allows to overcome the limitations of the standard ADS approach in terms of flexibility and performance. The numerical validation, carried out in the far-field and for narrow-band signals, points out that with affordable computational efforts it is possible to design planar array arrangements that outperform standard ADS-based designs as well as standard GA design approaches.