  We establish several relationships between the non-relativistic conformal
symmetries of Newton-Cartan geometry and the Schr\"odinger equation. In
particular we discuss the algebra $\mathfrak{sch}(d)$ of vector fields
conformally-preserving a flat Newton-Cartan spacetime, and we prove that its
curved generalisation generates the symmetry group of the covariant
Schr\"odinger equation coupled to a Newtonian potential and generalised
Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing
tensors and conformal Schr\"odinger-Killing tensors, and we discuss their
respective links to conserved quantities and to the higher symmetries of the
Schr\"odinger equation. Finally we consider the role of conformal symmetries in
Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic
vector fields on twistor space corresponds to the symmetry algebra
$\mathfrak{cnc}(3)$ on the Newton-Cartan spacetime.
