  This article is concerned with an extensive study of an infinite-dimensional
Lie algebra $\mathfrak{sv}$, introduced in the context of non-equilibrium
statistical physics, containing as subalgebras both the Lie algebra of
invariance of the free Schr\"odinger equation and the central charge-free
Virasoro algebra $Vect(S^1)$. We call $\mathfrak{sv}$ the
Schr\"odinger-Virasoro algebra. We choose to present $\mathfrak{sv}$ from a
Newtonian geometry point of view first, and then in connection with conformal
and Poisson geometry. We turn afterwards to its representation theory:
realizations as Lie symmetries of field equations, coadjoint representation,
coinduced representations in connection with Cartan's prolongation method
(yielding analogues of the tensor density modules for $Vect(S^1)$), and finally
Verma modules with a Kac determinant formula. We also present a detailed
cohomological study, providing in particular a classification of deformations
and central extensions; there appears a non-local cocycle.
