Wright first gained attention for her role as Kelly Capwell in the NBC Daytime soap opera Santa Barbara from 1984 to 1988.

She gained critical acclaim in her role as Jenny Curran in Forrest Gump (1994), receiving Golden Globe Award and Screen Actors Guild nominations for Best Supporting Actress.

In 1996, she starred in the lead role of the film adaptation of Daniel Defoe's Moll Flanders (1996), for which she received a Satellite Award nomination for Best Actress in a Drama.

She was nominated for a Screen Actors Guild Award for Best Actress for her role in She's So Lovely (1997), a film in which she co-starred with her then-husband Sean Penn.

Wright received her third Screen Actors Guild Award nomination for her role in the television film Empire Falls (2005).

[11] In 2017, for her performance in the fifth season, Wright was nominated for her fifth consecutive Primetime Emmy for Outstanding Lead Actress in a Drama Series.

In October 2017, she was the show's new lead for the final season, following the firing of Kevin Spacey due to sexual misconduct allegations.

For the latter, she became one of seven women to be nominated for the category six or more times for the same show, the first in 10 years since Mariska Hargitay for Law & Order: Special Victims Unit.

[18] Land premiered in January 2021 at the Sundance Film Festival to generally positive reviews, with specific praise towards the direction and the performances.

Peter Debruge in a review for Variety wrote: "So bless Wright for paring “Land” down to a beautiful haiku, and for delivering a performance that's ambiguous and understated in all the right ways," and "in a directorial debut so pure and simple it speaks to enormous self-confidence, has better instincts than to reveal outright.

"[21] Wright will next star in and produce Ben Young's thriller Where All Light Tends to Go, an adaptation of the book of the same name by David Joy.

[22] In April 2022, Wright joined the cast of Netflix's fantasy film Damsel directed by Juan Carlos Fresnadillo.

[23] Wright also starred in a Sony Pictures Classics and Miramax adaptation of Richard McGuire's Here in a Forrest Gump reteam with director Robert Zemeckis, actor Tom Hanks and writer Eric Roth coming on November 1st, 2024.

Wright was offered the role of Maid Marian in the film Robin Hood: Prince of Thieves, but turned it down because she was pregnant.

She is the narrator and executive producer of the documentary When Elephants Fight[47] which highlights how multinational mining corporations and politicians in the Democratic Republic of Congo threaten human rights, and perpetuate conflict in the region.