  We study affine Jacobi structures on an affine bundle $\pi:A\to M$, i.e.
Jacobi brackets that close on affine functions. We prove that there is a
one-to-one correspondence between affine Jacobi structures on $A$ and Lie
algebroid structures on the vector bundle $A^+=\bigcup_{p\in M}Aff(A_p,\R)$ of
affine functionals. Some examples and applications, also for the linear case,
are discussed. For a special type of affine Jacobi structures which are
canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures)
over a real vector space of finite dimension, we describe the leaves of its
characteristic foliation as the orbits of an affine representation. These
affine Jacobi structures can be viewed as an analog of the
Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real
finite-dimensional Lie algebra.
