Kučová qualified for the 2004 French Open, and drew 10th seed Vera Zvonareva in the first round of the clay-court tournament.

In 2008, Kučová competed at the 2008 GDF Suez Grand Prix in Budapest, Hungary.

[1] She knocked young Swiss player Timea Bacsinszky out of the tournament in the opening round, but failed to defeat hometown favourite Gréta Arn in the second, bowing out in straight sets.

Following 18 months absent from the tennis court, Kučová qualified for her first Grand Slam main-draw appearance in three years at the 2013 French Open.

There, she caused a huge upset by defeating 24th seed Julia Görges in straight sets.