Smith is most notably known for his two spells at the Ibrox club, as he led them to 21 titles in total, including 10 Scottish Premiership titles.
He also spent time as manager of Everton as well as the Scotland national team.
Rangers chairman Douglas Park said in a statement on Tuesday: "On behalf of the Rangers board of directors, staff and players, I convey my deep condolences to the Smith family. Walter leaves behind a wife, children and grandchildren, all of whom are in our thoughts and prayers at this difficult time.
"It is almost impossible to encapsulate what Walter meant to every one of us at Rangers. He embodied everything that a Ranger should be. His character and leadership was second to none, and will live long in the memory of everyone he worked with during his two terms as first team manager.
"I spoke with Walter as recently as last weekend. Even when he was battling illness, he was still able to provide advice and support. For that, I am personally grateful. I know that he continued also to maintain dialogue with senior members of staff, including our manager, Steven Gerrard.
"Walter will be sorely missed by all of us at Rangers.
"Winning 10 top-flight titles as manager, five Scottish Cups and six League Cups, as well as leading his club to the UEFA Cup final in 2008, he will be remembered by the football community across the world. His spells as Scotland manager as well as managing in the English Premier League underlined his credentials as one of the great modern day football managers.
"However, for Rangers supporters, he was much more than just a football manager. Walter was a friend to many, a leader, an ambassador and most of all -a legend."
