In this talk, we will discuss several aspects of Finsler spacetimes. First, we will discuss Ehlers-Pirani-Schild axioms and their compatibility with the Finsler setting. Then we will present some examples with certain issues of regularity and we will discuss to what extent these issues are important for singularity theorems. Finally, we will review the different approaches to obtaining the Finslerian Einstein equations and the role of the stress-energy tensor and the conservation of energy.