Livingstone, 24, is one of the most highly rated young batsmen in the country, and enjoyed a prolific season for Lancashire in 2017, averaging more than 50 in the County Championship with a match-winning double-hundred against Warwickshire at Old Trafford.
In addition, he made his international debut in June, and though he did not take his initial opportunities with scores of 16 and 0 in two T20s against South Africa, he is widely regarded as one of the obvious contenders for a middle-order berth in all formats for England.
After coming through the ranks at Lancashire, where he first played for the club aged 15, Livingstone stepped in as captain during the early part of the 2017 season when Croft was absent through injury.
"Liam has been seen as a potential leader as he has developed through the Lancashire Cricket system," Glen Chapple, Lancashire's head coach, said. "He showed maturity in the role, without it affecting his form when he captained the side earlier in the season, and he commands the respect of the dressing room."
"It is a huge honour to be named Lancashire captain," said Livingstone. "I feel privileged, humbled, and most of all, hugely excited by this opportunity.
"I thoroughly enjoyed captaining the side for a short period earlier this year and I can't wait to lead the Red Rose out again. The vision for the club is incredibly exciting and we've got a very talented group of players in the changing room."
The news, however, came as less of a thrill to Croft, who himself stepped in as leader in 2015 when Tom Smith was injured, and led the club to both that year's NatWest T20 title and promotion to the first division of the Championship.
"Being Captain of one of the biggest cricket clubs in the world has been amazing," said Croft. "It is something I wanted to continue to do, but I totally respect the decision that has been made. "Leading the Red Rose out onto the field for over 100 matches is something I have loved doing and will always cherish. "Next season I can fully concentrate on the playing side of my game which I'm really looking forward to. I wish Liam the very best of luck and I will always be here to support him and give my all for this great club."
Chapple paid tribute to his outgoing captain while unveiling his replacement. "In congratulating Liam on his appointment, I'd like to acknowledge Steven's leadership as captain since 2015," he said.
"He has led the side with purpose and character over the last three years and will continue to play a key role as a senior player in the future."