The Scottish Football Association described Smith as “one of the most successful managers in Scottish football history,” while Scotland’s First Minister Nicola Sturgeon called him a “true football great.”
“On behalf of the Rangers board of directors, staff and players, I convey my deep condolences to the Smith family,” Rangers chairman Douglas Park told the club’s website on Tuesday.
“Walter leaves behind a wife, children and grandchildren, all of whom are in our thoughts and prayers at this difficult time.”
As well as those 10 league titles, Rangers won five Scottish Cups and six League Cups with Smith as manager.
“It is almost impossible to encapsulate what Walter meant to every one of us at Rangers,” added Park. “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.
“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.”
Smith also managed English Premier League club Everton for four years and was also manager of the Scotland national team for two years.
“I looked up to Walter with so much respect,” West Ham’s Scottish manager David Moyes told the League Managers Association.
“He was such a great Scottish manager but, to me, he was an even better man,” added Moyes. “He was honourable, he showed great dignity, and always demonstrated such class.
“He was a true friend and mentor, and a man I feel privileged to have known. He will always be remembered fondly.”