Among the medical imaging modalities, ultrasound (US) imaging is one of the safest, most widespread, and least expensive method used in medical diagnosis. In the past decades, several technological advances enabled the advent of ultrafast US imaging, an acquisition technique capable of imaging large tissue zones at very high frame rates of multiple kilohertz. Achieving such frame rates on large tissue zones enables the analysis of complex physical phenomena occurring in the human body. For instance, such capability enables estimating both very fast and very slow flows occurring in the cardiovascular system, with high sensitivity. High frame rates also enable estimating micrometer tissue displacements induced by naturally occurring or externally induced shear waves that propagate through tissue at a few meters per second. Ultrafast US imaging is already at the origin of several breakthrough imaging modes such as shear-weave elastography and functional neuroimaging. One of the main advantages of pulse-echo US imaging is that it is a dynamic imaging modality. Conventional US images reconstructed using the well-known delay-and-sum algorithm are characterized by speckle patterns. Despite being an "illusion" of the imaging system, these patterns react coherently to underlying physical phenomena, thus containing positional information of the tissue being imaged that can be exploited by displacement estimation techniques. Because ultrafast acquisitions are performed using unfocused wavefronts, resulting images are of low quality, characterized by broad main lobes (low resolution) and high diffraction artifacts (low contrast) caused by grating lobes, side lobes, and edge waves. Such artifacts can be detrimental to both lesion detection and displacement estimation techniques, the latter being the core objectives of most ultrafast US imaging modes. A natural way of increasing the image quality of consecutive frames consists of averaging coherently multiple low-quality images obtained from differently steered unfocused wavefronts, at the expense of a reduced frame rate and possible motion artifacts. This thesis aims at answering the increasing need for US image reconstruction methods capable of producing high-quality images from single ultrafast acquisitions, may it be to improve the accuracy and robustness of ultrafast imaging modes such as shear-wave elastography, to reduce the cost and complexity of 3-D ultrasound scanners, or to mitigate the power and data transfer rate requirements of portable systems. This work builds in the context of inverse problems, with an efficient modeling of the physical measurement process (forward model) involved in ultrasound acquisitions. It leverages recent deep-learning-based projection methods to overcome a crucial limitation of regularized ultrasound imaging: conventional image-processing regularizers are not well suited to the high dynamic range and statistical properties of radio frequency ultrasound images, especially in the presence of speckle patterns. Physical modeling is fundamental to this work. It was crucial to derive a computationally tractable forward model for image reconstruction, but also to develop a highly efficient, spline-based, spatial impulse response ultrasound simulator that allowed generating sufficiently large datasets to train convolutional neural networks. Applications were carried out in single-plane-wave imaging, ultrafast displacement estimation, and sparse-array imaging.