  This is a pedagogical review of the subject of linear polymers on
deterministic finitely ramified fractals. For these, one can determine the
critical properties exactly by real-space renormalization group technique. We
show how this is used to determine the critical exponents of self-avoiding
walks on different fractals. The behavior of critical exponents for the
$n$-simplex lattice in the limit of large $n$ is determined. We study
self-avoiding walks when the fractal dimension of the underlying lattice is
just below 2. We then consider the case of linear polymers with attractive
interactions, which on some fractals leads to a collapse transition. The
fractals also provide a setting where the adsorption of a linear chain near on
attractive substrate surface and the zipping-unzipping transition of two
mutually interacting chains can be studied analytically. We also discuss
briefly the critical properties of branched polymers on fractals.
