Ukraine has claimed it still controls parts of Bakhmut after Russian forces said they had finally captured the besieged eastern city.

The conflicting claims follow a months-long slog in the city where Russian soldiers have had to grind for every inch of territory.

Kyiv has turned the battle -- for a largely insignificant city in the Donbas -- into a monumental struggle that has encapsulated the slow-moving and territorial nature of the ground war.

Who has claimed what?

Ukraine's deputy defense minister, Hanna Maliar, said Monday that Ukrainian forces were still in control of some buildings in the southwest of Bakhmut, two days after Russia claimed to have captured the city.

Maliar also claimed that Ukraine's troops were advancing on the city's flanks.

On Ukrainian television, she said: "Yesterday, the Ukrainian Armed Forces retained control of certain industrial facilities and private houses area in the southwestern area, the area where the aircraft [monument] is." The monument of a MiG-17 is in Druzhba Square in the southwest of Bakhmut. "Today, we still have control of this small part of the city. The fighting continues," she added.

Ukrainian President Volodymyr Zelensky had earlier said at the G7 summit in Japan: "We are keeping on, we are fighting."

That followed Russian claims to the contrary. On Saturday the chief of the mercenary Wagner group, Yevgeny Prigozhin, claimed to have captured Bakhmut after months of brutal fighting, saying he would hand it over to Russia later in May.

Russian President Vladimir Putin offered his congratulations for "the completion of the operation to liberate Artemovsk," Russian state news agency TASS reported the Kremlin as saying, using the Soviet-Russian name for Bakhmut.

How important is Bakhmut?

Bakhmut sits toward the northeast of the Donetsk region, about 13 miles from the Luhansk region, and has long been a target for Russian forces. Since last summer the city has been a stone's throw from the front lines.

Donbas -- the vast, industrial expanse of land in Ukraine's east, encompassing the Luhansk and Donetsk regions -- has been the primary focus of Russia's war effort since last spring, after its initial lunge toward Kyiv and central Ukraine failed.

The battle has been compared to the kind of fighting seen in World War I, with soldiers fighting in a hellish landscape of mud and trenches, trees and buildings mangled by artillery fire.

While Russian forces have continued their slow street-by-street advance in Bakhmut for many months, over the past two weeks Ukrainian forces have managed to re-capture small pockets of territory held by Russian troops to the northwest and southwest of the city.

Russian forces, bolstered by members of the Wagner mercenary group, have taken heavy losses trying to capture the city.

There are no official casualty figures, but earlier this year a NATO source told CNN they estimated that for every Ukrainian soldier killed defending Bakhmut, Russia lost five.

A 'meat grinder' of a battle

The battle has also highlighted an extraordinary rift among Russian forces, with Prigozhin at one point accusing a Russian brigade of abandoning its position in the city and railing several times at the Defense Ministry over a lack of ammunition.

Prigozhin, a former catering boss who has grown in prominence throughout the war, compared the battlefield to a "meat grinder."

Bakhmut's fall would be an undoubted boost to Prigozhin, who recently announced his men would pull out entirely because dwindling ammunition supplies and mounting losses meant there was "nothing left to grind the meat with."

Over the early part of 2023, the routes into Bakhmut had gradually come under the control of Russian forces and the battle for the city turned into an inch-by-inch grind, with Ukrainian forces repelling dozens of assaults each day.

Only two roads out to the west have remained outside Russian control, though for Ukrainian forces trying to re-supply the city, the drives in have been treacherous.

Before they began their slow push through the city center, Wagner troops first sought to encircle the city in a wide arc from the north.

In January they claimed the nearby town of Soledar, and later took a string of villages and hamlets north of Bakhmut, making Ukraine's defense of the city increasingly perilous.

But even as Moscow's troops closed in and most residents fled through dangerous evacuation corridors, a small group of Ukrainian civilians remained in the ruined city. Before the war, around 70,000 people lived in Bakhmut, a city once famous for its sparkling wine.

As of March, the population stood at less than 4,000 and most of the once thriving city has been reduced to ashes and rubble.

In his comments at the G7, Zelensky said pictures of ruined Hiroshima he has seen on his visit to the Japanese city "really remind" him of Bakhmut.