All-dielectric metasurfaces have been intensively researched as a low-loss, flat-optics platform for the advanced manipulation of electromagnetic wave propagation. Among the numerous metasurface-enabled functionalities, particular focus has been recently placed on the engineering of components with an extremely narrowband response, stemming from so-called bound states in the continuum, which can boost the performance of, among others, non-linear, sensing, or lasing devices thanks to the enhanced lightmatter interaction and strong field enhancement. On the other extreme in terms of their operating bandwidth, spatially modulated gradient metasurfaces have opened the path towards ultrabroadband, achromatic, flat components, which are key to applications such as imaging, holography, and the processing of ultra-short pulses. In this work, we provide a critical overview of recent developments in both fields, highlighting the underlying physical concepts, reporting their experimental demonstration in a broad range of applications with unprecedented performance, and providing a future outlook towards metasurfaces with extreme spectral responses as the enabling element in emerging applications. We provide a critical overview of recent advances in all-dielectric, strongly resonant and gradient metasurfaces, as their performance is pushed to the extreme in view of emerging flat-optics applications.