Single-walled carbon nanotubes (SWNTs) are nano-cylinders obtained by wrapping one layer of graphene; due to their very high aspect ratio, they are the prototypical quantum confined one-dimensional systems. The unique mechanical, electronic and optical properties of SWNTs open up transversal application possibilities in many fields of science and technology, with particular emphasis on optoelectronics and photonics. A prerequisite for many of these applications is a thorough understanding of the nature and dynamics of their elementary excitations. This review aims at summarizing the current understanding of the ultrafast photophysics of SWNTs, based on two decades of experimental investigations. After discussing the morphological and electronic properties of SWNTs and introducing the different photogenerated species, we will briefly describe the ultrafast spectroscopic techniques most commonly used for their characterization. We will then present the experimental evidence that has led to establish the nature (singlet and triplet excitons, biexcitons, trions and free charges) and the relaxation pathways of photoexcitations in SWNTs.