Bands of Ukrainian soldiers fighting to take back territory on the eastern bank of the Dnipro River, an area long controlled by Russia, have been bombed by Russian warplanes, assaulted by Russian infantry and stalked by drones.

Still, battered and outgunned, the Ukrainian forces have managed to hold onto a handful of positions across the river for more than a month and are expanding their assaults on Russian forces there to target their vital supply lines.

The ultimate objectives of the Ukrainian campaign remain unclear: Is it aimed mainly at unbalancing Russian forces — using limited assaults to force the Kremlin to move troops to the area, hoping to create weaknesses along other parts of the front? Or does Ukraine have more ambitious objectives, like trying to mount a major cross-river assault aimed at taking back a substantial amount of territory and dramatically reshaping a front line that has barely moved in a year?

Many Western military analysts have voiced skepticism that Ukraine can establish the kind of bridgehead that would allow its forces to move artillery and heavy armor across the river, which they would need to carry out large-scale offensive operations.

Still, the sustained attacks could prove difficult for Russia, especially if Ukraine can interfere with critical Russian supply lines. Whatever the Ukrainian intentions, the marshy wetlands along the Dnipro are simmering.

Here is a brief look at how the fighting has evolved, where things stand, and the risks and rewards should Ukraine attempt the most ambitious battlefield river crossing since World War II.

Much of the current state of fighting remains shrouded in secrecy and is deliberately obfuscated by both sides.

But military analysts using geolocated combat footage confirmed last month that Ukrainian forces are holding onto several footholds and are engaged in clashes in a string of villages stretching from Oleshky, opposite the city of Kherson, to Korsunka, a town about 30 miles up river.

The commander of a special Ukrainian unit fighting on the east bank said his soldiers had made their first forays across the river in August.

In late October, Ukrainian marines joined the fight, and in mid-November the marines announced that they were holding several bridgeheads. It was at that point that President Volodymyr Zelensky mentioned the operation for the first time.

As the Ukrainian assaults across the river intensified, so, too, did Russia’s response.

In late October, Russian warplanes started blanketing the area with 500- and 1,000-pound bombs and used TOS-1A thermobaric artillery systems, which suck in oxygen from the surrounding air, to devastating effect, according to soldiers and combat footage.

By attacking Russian forces on the east bank of the river, Ukraine is forcing Russia to move forces from other parts of the front, according to Russian military bloggers, the Ukrainian military, British military intelligence and military analysts.

But the fighting is taking a heavy toll on Ukrainian forces, with soldiers releasing combat footage of fierce fighting and harsh living conditions.

Ukraine appears willing to risk exposing some of its best fighters to such a precarious and difficult fight because the rewards of a successful operation could be transformative.

If Ukraine is successful in establishing enduring positions across the river, its forces would be within 30 miles of Crimea — putting a vital transit hub on the peninsula in range of Ukrainian artillery, reshaping the geography of the battlefield, and making it even harder for Moscow to bring food, fuel and ammunition to tens of thousands of soldiers over the winter.

Yevhen Dykyi, the former commander of the Ukrainian Aidar battalion, said Ukrainian troops were “closing in on” a critical highway connecting Crimea to Melitopol — an essential artery in the Russian supply chain.

“The next task is more difficult,” he said last week on Ukrainian television. “In particular, to expand this foothold, break through the Russian defense and gain operational space.”

A chorus of prominent Russian military bloggers have criticized Russian commanders for not taking the threat from Ukraine seriously enough.

As reports of increased Ukrainian activity grew in October, the Kremlin replaced the commander in the area, Col. Gen. Oleg Makarevich, with Col. Gen. Mikhail Teplinsky, who had previously served as head of Russia’s elite Airborne Forces.

The Institute for the Study of War, a Washington-based think tank, said in a report last month that Russia’s military “will likely struggle to redeploy combat-effective reinforcements” to the area while also engaged in defensive operations in the Zaporizhia region, to the northwest, and sustaining other offensive efforts in eastern Ukraine.

The Kremlin’s main response has been to use its dominance in the air to carpet-bomb the areas where they believe the Ukrainians have footholds, hoping that the withering bombardments will dislodge them. Recently released Russian and Ukrainian drone footage reveals once peaceful riverside villages now razed to the ground, without a single building standing.

Several prominent Russian military bloggers have reported midlevel command problems, with Russian soldiers posting videos complaining about being ordered to go on suicidal missions while living in tough conditions.

To expand their tenuous hold on the Dnipro’s eastern bank, the Ukrainians need to find reliable ways to get supplies and reinforcements across the river — no easy task.

“A river crossing under fire is one of the most difficult operations in land warfare,” said John D. Hosler, a professor of military history at the Command and General Staff College at Fort Leavenworth, Kan. “Imagine an hourglass, in which the sand flows from one large container through a narrow channel into another: River crossings are the horizontal expression of the same.”

Soldiers and equipment are vulnerable at every stage of the operation: when they mass to prepare for the crossing, as they move across the “wet gap,” and once again on the far side.

While the Dnipro River narrows as it passes the port city of Kherson, and Ukraine has combat-tested engineering units — as well as bridging equipment designed for the task — it would be hard to move large quantities of material across the river without being detected.

Widespread use of drones has made an already treacherous undertaking more deadly. Once across the river, the marshy flatlands on its eastern bank offer little natural cover.

Beyond the possible operational benefits for Ukraine that could come from expanding the area under its control along the river, a successful crossing effort would also likely raise morale sharply, particularly after a year of toil and bloodshed but little advancement on the ground.

But a failed campaign would mean that more of the country’s best soldiers are lost.

No modern army has attempted anything even close to this scale under these conditions since World War II, and historians said it might be better to look back further for an analogy: George Washington leading his soldiers across the Delaware in December 1776.

“Washington’s audaciousness ended up being worth the risk: It not only gained him a victory at Trenton, but also boosted the morale of his own beleaguered forces,” Mr. Hosler said. That war would grind on until 1783, but the battlefield victory gave the struggling Continental Army something it desperately needed at that moment: hope.

Anna Lukinova contributed reporting.