  In this work, we study the higher differentiability of solutions to the
inhomogeneous fractional $p$-Laplace equation under different regularity
assumptions on the data. In the superquadratic case, we extend and sharpen
several previous results, while in the subquadratic regime our results
constitute completely novel developments even in the homogeneous case. In
particular, in the local limit our results are consistent with well-known
higher differentiability results for the standard inhomogeneous $p$-Laplace
equation. All of our main results remain valid in the vectorial context of
fractional $p$-Laplace systems.
