The objective of the talk is to present the different techniques and results that have been achieved in the past years towards finding short vectors in ideal and module lattices. We should (hopefully) cover the following results: Campbell-Groves-Shepherd'14, Biasse-Song'16, Cramer-Ducas-Peikert-Regev'16, Cramer-Ducas-Wesolowski'17, Pellet-Mary-Hanrot-Stehlé'19, Ducas-Plançon-Wesolowski'19, Lee-Pellet-Mary-Stehlé-Wallet'19, Mukherjee-Stephens-Davidowitz'19, Kirchner-Espitau-Fouque'19.