In the semiconductor industry, optical lithography is presently the most widespread technology used to print a geometrical pattern on a semiconductor wafer. Because of the plans imposed by the International Technology Roadmap for Semiconductors (ITRS) for more powerful and smaller chips, new printing technologies capable of printing smaller features on the wafer need to be developed. Extreme UltraViolet Lithography (EUVL) is a promising candidate for the next-generation of pattern technology beyond the current machines that operate at wavelength of 193 nm based on optical lithography. EUVL uses photons of 13.5 nm wavelength to carry-out the imaging and therefore it is suitable for addressing not only the 22 nm half-pitch nodes but also several nodes beyond that. However, material properties at the EUV range make this technique completely different from present-day lithography. EUV radiation is strongly absorbed in all materials and gases and therefore multilayer (ML) aspherical mirrors have to be used in the projection optics. Moreover, because of the partial reflectivity of a ML mirror (about 70%), part of the radiation is absorbed by the mirrors causing thermally induced aberrations in the optical projection box. Adaptive optics (AO) is an attractive technology that could be implemented in lithography machines as a method to compensate these aberrations and achieve high quality diffraction-limited imaging. Since the aberrations we aim to correct are very small, a significant amount of research has been done to develop metrology techniques that can measure them with extreme accuracy and in high speed. Most accurate wavefront sensing are usually done using interferometry techniques, but these require extremely stable setups and laborious measurement interpretations. In this thesis, we focus alternatively on another technique, called phase retrieval, which can be a valid substitute for interferometry. In the first part of this thesis, we show how phase retrieval can be employed as a method to increase the accuracy of an already established wavefront measuring technique based on an Hartmann wavefront sensor (HWS). The HWS technique relies on the assumption that the wavefront slope can be locally approximated by a linear function which is sensed on each sub-aperture of the sensor. Departures from this linearity assumption is the main limiting factor in the accuracy of the reconstructed wavefront. The implementation of the phase retrieval method presented here aims to recover the nonlinearities in the phase distribution and therefore achieves an improved reconstruction. Simulations showed that the RMS wavefront reconstruction has an accuracy of the order of 10?4?. This value is typically one order of magnitude lower than the linear slope computational method. Experiments have been carried-out to compare the retrieved results with an independent wavefront measurement showing good agreement with the theoretical predictions. In the second part of this thesis, we focus our attention on phase retrieval from the intensity distributions of the field in the focal region. This method has already been demonstrated to be capable to estimate the phase aberrations of an optical system. However, the conventional implementation requires the measurement of intensity distributions in several different planes in the focal region in order to obtain a stable phase estimation. As an alternative, we develop here a method where phase retrieval from a single plane of measurement is achieved. Hence, our task is to reduce the measurement time and the amount of data to be analysed. As a result, we show that using a statistical approach, an optimal plane of measurement at one focus depth according to the axial Rayleigh criteria, can be identified. The analysis of the statistical uncertainties and correlation coefficients shows that the phase retrieval from a single intensity measurement plane at said out-of-focus distance leads to a stable phase estimation with minimum uncertainties and minimum RMS wavefront deviation. The concept of phase retrieval from a single intensity measurement plane is further investigated. By expanding the exponential term in the exit pupil function in Taylor series up to the first order we are able to describe, with proper approximations, the contribution of the aberration functions to the intensity distribution in the focal region. This allows us to show that aberration functions characterised by an even spatial distribution do not contribute to the intensity distribution of the field in the Gaussian focal plane. Hence, the problem of retrieving phase information from intensity measurements in such a plane is impossible. Moreover we show that, also in this approximation, it is possible to identify an optimal plane of measurement. By looking at the contribution of the aberrations to the intensity distribution as a function of the out-of-focus distance, we have found analogous results. Namely, the aberrations contribute to the intensity distribution in a minimal manner for distances close to the focal plane. Conversely, the contribution is maximised for a distance of one focus depth (Rayleigh criteria), confirming the previous findings. To exploit these results, a new experimental AO system based on the phase retrieval method has been built. In particular, this has been possible by joining the knowledge of the optimised phase retrieval method together with an innovative type of control strategy for the AO system. These two key elements need to work in perfect symbiosis: phase retrieval has to provide the accurate measurement of the phase, and the control strategy needs to translate the phase information to an electrical signal that drives the compensating element (e.g., a deformable mirror) in a correct and efficient way. For this reason, we also provide a complete analysis of the control algorithm employed in our setup, focusing on the convergence speed, shape quality and voltage saturation. This allows us to demonstrate that aberrations can be corrected in real-time up to ?/100 with a simple optical bench and a standard computer. Finally, the last chapter of this thesis is dedicated to the feasibility analysis of the phase retrieval method implemented on a patented EUVL optical design. Thermally induced phase aberrations are modelled in the system on the basis of a thermo-elastic Finite Element Model (FEM) simulation. A metrology procedure using different operational wavelengths i.e., 13.5 nm 193 nm and 633 nm has been proposed. Simulations have shown that the RMS wavefront reconstruction scales in general with the wavelength. Nevertheless, wavefront metrology at 193 nm has shown an accuracy below 650 pm RMS wavefront. This result is sufficient to fulfil the requirements and moreover allows relaxed experimental conditions for the metrology procedure.