  We discuss non-relativistic variants of four-dimensional ${\cal N}$=4
super-Yang-Mills theory obtained from generalised Newton-Cartan geometric
limits of D3-branes in ten-dimensional spacetime. We argue that the natural
interpretation of these limits is that they correspond to non-relativistic
D1-branes or D3-branes intersecting the original D3-branes. The resulting gauge
theories have dynamics that reduce to quantum mechanics on monopole moduli
space or two-dimensional sigma-models on Hitchin moduli space respectively. We
show that these theories possess interesting infinite-dimensional symmetries
and we discuss the dual $AdS$ geometries.
