Excitons, the bound states of an electron and a hole in a solid material, play a key role in the optical properties of insulators and semiconductors. Here, we report the observation of excitons in bilayer graphene (BLG) using photocurrent spectroscopy of high-quality BLG encapsulated in hexagonal boron nitride. We observed two prominent excitonic resonances with narrow line widths that are tunable from the mid-infrared to the terahertz range. These excitons obey optical selection rules distinct from those in conventional semiconductors and feature an electron pseudospin winding number of 2. An external magnetic field induces a large splitting of the valley excitons, corresponding to a g-factor of about 20. These findings open up opportunities to explore exciton physics with pseudospin texture in electrically tunable graphene systems.