  This review-type paper is based on a talk given at the conference {\'E}tats
de la Recherche en M{\'e}canique statistique, which took place at IHP in Paris
(December 10-14, 2018). We revisit old results from the 80's about one
dimensional long-range polynomially decaying Ising models (often called Dyson
models in dimension one) and describe more recent results about interface
fluctuations and interface states in dimensions one and two.
