We shall review the remarkable properties of the O(1) Temperley--Lieb loop model and how they led Razumov and Stroganov to formulate a conjecture which relate it to Fully Packed Loops (FPL) and Alternating Sign Matrices (ASM). We shall then discuss recent attempts at proving this conjecture: in particular, we shall try to generalize it by introducing inhomogeneities and explain the connection with the Izergin--Korepin/Okada determinant formulae for the six-vertex model.