In the MCU (movies)

In Avengers: Age Of Ultron it seemed that she and Bruce Banner had grown close and have feelings towards each other. Despite the sudden romantic relationship between the two not being addressed in any previous movies or shots.

It is unclear whether the Russo Brothers will continue with that particular story line or not. It is hinted that the Winter Soldier will be have a conversation with her, but it is unclear.

Also it seemed in the film "Avengers Confidential: Black Widow & Punisher" Natasha had a relationship with scientist Elias Star who ended up being a bad guy.

In the Comics

Natasha Romanoff,in the comics had a relationship with Mathew Murdock aka. Daredevil. Also Clint Barton, Bucky Barnes and Steve Rogers where they end up having a child named James Rogers who is featured in the animated movies.

The relationship between Bucky and Natasha happened when Nat was in the Red Room, Bucky was to train her and they soon fell in love although both Red Room and Hydra forbidding it so they brainwashed Bucky so he couldn't remember. They got back together later on.

Steve and Natasha also had a relationship and they had a child named James (named after Bucky) in the comics, in the animated movie "Next Avengers: Heroes of Tomorrow". It is also evident that they have a relationship in "Ultimate Avengers" and "Ultimate Avengers 2"

Hope that answers the question, drop a question in the comments. :)