In this talk I will give "affine" generalizations of the notions of semistandard Young tableaux and the RSK (Robinson-Schensted Knuth) algorithm. The affine tableaux come in two versions, a weak and a strong one. Weak and strong tableaux are certain chains in the weak and strong (Bruhat) orders of the affine symmetric group respectively. Our main result is an affine insertion algorithm, generalizing the RSK algorithm, which sends matrices to pairs (P,Q) of a strong tableau P and a weak tableau Q of the same shape. While this work is geometrically motivated, no knowledge of geometry will be assumed. This is joint work with Luc Lapointe, Jennifer Morse and Mark Shimozono. I will also explain an extension of part of this setup to the general Kac-Moody setting, using the framework of Fomin's dual graded graphs. This last part is joint work with Mark Shimozno.