Whereas pulse-echo ultrasound imaging relied on focused acoustic waves since its inception, the last two decades have seen the development of techniques based on unfocused waves, including ultrafast ultrasound imaging. In large part due to the emergence of the latter, new methods able to image quantitative biomarkers have also been proposed, a promising example being local tissue speed of sound. In addition, the recent years have witnessed a gradual shift from hardware to software processing of ultrasound data, along with the birth of portable and ultra-portable transducers. These factors highlight the potential of complex computational methods to improve ultrasound imaging quality while reducing the amount of data required. In particular, we study in this thesis the use of inverse problems in ultrafast ultrasound for both qualitative and quantitative imaging. Local speed-of-sound variations in the medium are overlooked by traditional ultrasound image reconstruction methods such as delay-and-sum. Phase aberrations are thus introduced in reconstructed images, ultimately affecting diagnosis with ultrafast ultrasound. To circumvent this issue, we first develop a theoretical framework where the measured ultrasound echo signals are expressed in a local plane-wave basis. Based on this framework, we devise an adaptive image reconstruction method based on an inverse problem to correct for phase aberrations. Better recovery of fine structure and better contrast are therefore achieved with respect to delay-and-sum and the recently introduced SVD beamformer. Besides being a source of aberrations, the local variations of tissue speed-of-sound can provide a quantitative biomarker for certain conditions like non-alcoholic fatty liver diseases. However, current pulse-echo speed-of-sound imaging methods often lack robustness. A major contribution of this thesis is to propose a local speed-of-sound estimation method based on the local angular basis we already relied on to correct phase aberrations. We show that the proposed method improves the stability of speed-of-sound estimation with respect to a state-of-the-art method, especially when the amount of data acquired is small. In addition, we propose a new regularization strategy for speed-of-sound imaging factoring in information from a traditional B-mode image, ultimately improving imaging accuracy with respect to previous approaches. Finally, we present an operator designed to take into account refraction effects into the inverse problems involved in pulse-echo speed-of-sound imaging. For the last major contribution of this thesis we disregard speed-of-sound variations and propose a statistical model of data acquisition in pulse-echo ultrasound. We first quantify the covariance matrix of the measured echo signals. We then propose two inverse problems to estimate from measured echo signals two types of images usually obtained by delay-and-sum and post-processing, namely despeckled and deconvolved images. We show that the despeckled images obtained with the proposed method are nearly free of diffraction artifacts even if they are reconstructed using a single insonification. Regarding the deconvolved images, we show that the proposed approach manage to consistently reduce artifacts and enhance contrast with respect to a previous approach.