The first passage percolation (FPP) model is a time evolution version of the bond percolation model: each edge in a cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region B(t), which consists of those vertices that can be reached from the origin within a time t > 0. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region B(t)/t converges to some limit shape. In this talk, I will give a general FPP model defined on crystal lattices, and show a general version of the shape theorem. I will also show the monotonicity of the limit shapes under covering maps, thereby providing insight into the limit shape of the cubic FPP model. This talk is based on the preprint https://arxiv.org/abs/2009.11679.