Quite extensively.
The first US combat mission against Italian forces took place on June 21, 1942, when US bombers deployed to British airbases in Egypt struck the harbor at Benghazi, then part of Italian Libya. Eight days later, a team of Italian-American OSS agents was landed on Sardinia to coordinate with anti-Fascist partisans; all five were captured the next day and executed. These minor probes were soon overtaken by much larger campaigns.
EDIT: see responses below, the agents were imprisoned, not executed. Stupid old brain...
On November 8, 1942, British and American forces invaded French North Africa. By the end of the month, Allied forces controlled Morocco and Algeria, and in January Field Marshal Montgomery broke Axis lines in Libya, forcing the Germans and Italians into a redoubt in Tunisia. Over the next three months, thousands of Italian troops were forced to surrender.
(German forces in Tunisia were better-motivated and better-led, but the Italian armed forces were not the lackluster mob of general belief; individual units fought bravely and even viciously on many occasions, although they often suffered from logistical problems, poor combined-arms support, and especially poor training.)
Not only did the losses in East Africa and North Africa deprive Italy of hundreds of thousands of soldiers, but the campaign to hold Tunisia cost Italy most of its merchant marine and navy.
On June 11, 1943, British paratroopers seized the small islands of Pantelleria and Lampedusa without a fight; this indicated clearly that the Italian military was a spent force and the final approval for Operation Husky was issued. On July 11, British and American paratroopers landed; around a thousand died, half in a horrifying friendly fire incident where beachhead troops opened fire on approaching transport planes. Despite this disaster, within two days 80,000 Allied troops were ashore, facing three times that number of Italians and two German divisions.
General George S. Patton led the charge north; on July 22, elements of his army reached Palermo on the northern coast of Sicily. This attack cost the Allies fewer than 300 casualties, while inflicting around 2,000 and accepting the surrender of over 50,000 Italian troops. While Patton was rolling up Sicily, American bombers struck Rome for the first time, killing 3,000. Italy's King Victor Emmanuel had had enough, and on July 25 he ordered Benito Mussolini to resign as prime minister. Field Marshal Pietro Badoglio succeeded him, assuring the Germans Italy would honor its alliance while also reaching out to the Allies. Germany rightly took little stock in these promises and withdrew its 40,000 troops from Sicily; the Allies did little to contest this withdrawal, unwilling to risk heavy casualties by taking on German fortifications around the Strait of Messina.
The Sicily campaign cost the Allies 5,000 dead, 17,000 wounded, and 22,000 hospitalized for malaria. The Axis lost 9,000 dead, 20,000 wounded, and 140,000 captured. The fall of Sicily also opened the Mediterranean to Allied shipping, allowing a flood of supplies to reach the Soviet Union.
Badoglio's government offered a surrender but dithered on a final date, cowed by the growing German presence in mainland Italy; the Allies punctured this by announcing they had accepted Italy's surrender on September 8. It took almost a month for the Allies to force their way north to Naples against stiffening German resistance. On September 12, German commandos rescued Mussolini from imprisonment and set up a puppet Italian government in the north. 600,000 Italians, many of them former conscripts and others considered politically suspect, were shipped north to Germany where they spent the war as conscript laborers.
Mussolini's new puppet Social Republic of Italy conscripted a new army and police force, fielding several hundred thousand soldiers and police officers, but most of its motivated units were tasked with fighting anti-Fascist partisans, rather than the Allies. Despite this, Italian forces were deployed to reinforce the German positions during the grinding campaign of attrition that took up the next two years; the final toll was 50,000 Allied dead and 150,000 wounded, 50,000 German dead and 450,000 wounded or captured - and almost 90,000 of the Social Republic's soldiers were killed, as were over 100,000 Italian civilians. The German lines in Italy finally crumbled in April 1945, and Mussolini himself was captured and killed by Italian partisans on April 28, 1945, marking the effective end of the war in Italy.
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