Pkdgrav3 is an 𝒪(N) gravity calculation method; it uses a binary tree algorithm with fifth order fast multipole expansion of the gravitational potential, using cell-cell interactions. Periodic boundaries conditions require very little data movement and allow a high degree of parallelism; the code includes GPU acceleration for all force calculations, leading to a significant speed-up with respect to previous versions (ascl:1305.005). Pkdgrav3 also has a sophisticated time-stepping criterion based on an estimation of the local dynamical time.