Drama:
* The Sopranos (1999-2007)
* Sex and the City (1998-2004)
* The Wire (2002-2008)
* Deadwood (2004-2006)
* Rome (2005-2007)
* Entourage (2004-2011)
* Six Feet Under (2001-2005)
* Oz (1997-2003)
* Carnivàle (2003-2005)
* Big Love (2006-2011)
* Boardwalk Empire (2010-2014)
* Game of Thrones (2011-2019)
* True Blood (2008-2014)
* Treme (2010-2013)
* The Newsroom (2012-2014)
* Veep (2012-2019)
* True Detective (2014-present)
* Silicon Valley (2014-2019)
* Westworld (2016-present)
* Succession (2018-present)
* Euphoria (2019-present)
* The White Lotus (2021-present)
* House of the Dragon (2022-present)
Comedy:
* Curb Your Enthusiasm (2000-present)
* Sex and the City (1998-2004)
* Entourage (2004-2011)
* Veep (2012-2019)
* Silicon Valley (2014-2019)
* Barry (2018-present)
* Insecure (2016-2021)
* Hacks (2021-present)
Other:
* Band of Brothers (2001)
* The Pacific (2010)
* John Adams (2008)
* Mildred Pierce (2011)
* Generation Kill (2008)
* The Deuce (2017-2019)
* Watchmen (2019)
This is not an exhaustive list, and many other excellent shows have aired on HBO.
Note: Some shows have multiple seasons, and the years listed indicate the initial run. Many series are also available on streaming services like HBO Max.