Delaney was raised in Marblehead, Massachusetts. He attended New York University's Tisch School of the Arts and graduated with a degree in musical theater in 1999.

Delaney has publicly shared his experience with clinical depression and alcohol addiction. In 2002, Delaney blacked out while driving and drove his car into a building owned by the Los Angeles Department of Water and Power. In the wreck, he broke his left wrist and right arm, and had both of his knees scratched to the bone. This prompted Delaney to stop drinking, and he has been sober since February 4, 2002.

Delaney first came to the attention of the public via Twitter, on which he started posting in 2009. By 2016, Delaney had over 1.2 million followers. While other comics were hesitant to share their material on social media, Delaney is considered one of the first comedians to use social media to publish jokes. Delaney credits comedy Writer and Director Graham Linehan with his rise in popularity after Linehan began responding to Delaney's tweets.

In 2010, Paste magazine named Delaney one of the ten funniest people on Twitter. In May 2012, he became the first Comedian to win the 'Funniest Person on Twitter Award' at The Comedy Awards hosted by Comedy Central.

In December 2011, Comedy Central announced that Delaney would shoot a pilot for a variety show, called "@RobDelaney". However, the series was never picked up by them.

His book Rob Delaney: Mother. Wife. Sister. Human. Warrior. Falcon. Yardstick. Turban. Cabbage. was published by Spiegel & Grau in November 2013.

Since moving to the UK in 2014, Delaney has appeared on several British panel shows, including Have I Got News for You, Would I Lie to You?, 8 Out of 10 Cats Does Countdown, The Big Fat Quiz of the Year and Room 101.

Instead, Delaney moved on to co-write and co-star with Sharon Horgan in Catastrophe, that began airing in the UK on 19 January 2015. The show follows them after they get together following a brief affair while he is in the UK on Business, and then moves there permanently after he learns that she has become pregnant. It debuted in the United States on Amazon in June, 2015. There are now three seasons on Amazon and the show has been renewed for a fourth.

Delaney is a left-wing progressive and often writes about political issues. He claims to choose presidential candidates based mostly on their policies regarding education and health care, reasoning: "Make it easier for your citizens to be healthy and smart and they will save you in ways you have yet to imagine." He supported Bernie Sanders in the 2016 Democratic primary and voted for Hillary Clinton in the subsequent general election. After Donald Trump was elected President, Delaney joined the Democratic Socialists of America, the National Association for the Advancement of Colored People, and the American Civil Liberties Union.

On June 15, 2017, Rob Delaney appeared as a panelist on the BBC's Question Time, chaired by David Dimbleby, in Coventry, England.

Delaney is married; he and his wife have two sons, having lost their third son in January 2018. In February 2018, Delaney publicly shared that their son Henry had died, age two and a half, the month before, after having been diagnosed with a brain tumor in 2016 and treated extensively. Delaney and his family live in London.