Costolo, whose departure was announced after the markets closed Thursday, will be replaced in the short term by Chairman Jack Dorsey, a Twitter co-founder who also serves as CEO of payment company Square.
Twitter said on Thursday that it will consider candidates both from within and outside of the company for the permanent CEO role. Costolo won't receive any severance pay as he steps down.
Shares of the company took off on the news in after-hours trading, heading as high as $39.29 from the closing price of $35.83. In premarket trading on Friday, shares were up 2.65% to $36.78.
Twitter reaffirmed its second-quarter guidance, including revenue of between $470 million and $485 million and EBITDA of between $97 million and $102 million.
As CEO, Costolo's approval rating among Twitter employees had been slipping, down to 87% this year from 94% in 2014, according to a survey conducted by workplace research Web site Glassdoor. That rating was still above the average of 69% for CEOs, Glassdoor said.
While there were no solid predictions, analysts acknowledged the company is at a crossroads. Some suggested that a new chief executive might offer the opportunity to take the company in a new direction, while others wondered whether Costolo's departure would clear the way for Twitter's acquisition.
Here's what a few had to say: