Last week at the Jazz Standard, saxophonist Stefano di Battista extracted much comic mileage by introducing his French pianist, Eric Legnini, and his Sicilian bassist, Rosario Bonaccorso, by saying that they were from Detroit, while announcing that his drummer, the Lincoln Center Jazz Orchestra’s Herlin Riley, was from Palermo. Most foreign jazz musicians regard it as an honor when you can’t tell by listening that they were born in Vienna or Rome rather than Chicago or Memphis, so in a way it was the highest praise he could give.

Yet there is a real tradition of Italian (and especially Italian-American) jazz. Mr. di Battista made this point himself when he quoted the Neapolitan standard “Arrivederci Roma” in the coda to “Laura.” Sixty years ago, Louis Prima, a musician generally remembered more as an irreverent funster than a serious innovator, achieved the remarkable feat of introducing the tarantella, the traditional Neapolitan dance form (Italy’s answer to the polka or the rumba) into American jazz and pop.

In the 1940s and 1950s, there was even an entire school of Italian-American tenor players, such as Flip Phillips, Charlie Ventura, and Vido Musso, whose approach was somewhere between Coleman Hawkins and Enrico Caruso: They were strong, muscular soloists with a big tone and a flair for the dramatic. Musso especially made a strong paesano statement by choosing “Sorrento” as his feature with Stan Kenton’s orchestra. More recently, the extraordinary tenor saxophonist Joe Lovano featured himself in an album-length tribute to Caruso.

A monstrously prodigious alto and soprano virtuoso, Mr. di Battista is the best known of younger Italian jazz musicians, the only one who records for a major American label, and one of the few who has enough star power to appear regularly in America. But he’s far from the only one worth listening to. This week New York’s jazz venues will see a veritable Italian invasion: “Jazzitaliano in New York,” a festival that runs throughout this week at over a half dozen different venues, showcases several dozen musicians in six different bands. Ira Gitler, who has been covering jazz since the dawn of the modern era 60 years ago, tells me that today’s Italian jazz performers are the best in Europe. I haven’t traveled widely enough to either dispute or substantiate that judgment, but between this week’s slate of performers and several recent releases on CD, I don’t have to.

Mr. di Battista’s appearances last week were not timed to coincide with the festival but the release of his fourth Blue Note album, “Parker’s Mood” (66740).Mr. di Battista (like Bird, an alto saxophonist) is one of those high-powered players – I class Chris Potter, James Carter, and Bobby Watson in this category, as well – who would have instantly been offered a job in The Jazz Messengers 20 years ago. He plays with great aggression and soul – the Italians must have some word for it.

Other contemporary Italian players bring the traditions of Italian folk and pop music into a jazz context. This is shown by two CDs released last year, Quadro Nuevo’s “Canzone Della Strada” (Justin Time 8504) and Enrico Pieranunzi’s “FelliniJazz” (CamJazz 5002).

Quadro Nuevo splits the difference between the two continents rather evenly. They play ballads and tarantellas – “songs of the street” according to my pig Latin Italian – in a quartet with a front line of accordion and reeds. Mulo Francel alternates between three different clarinets and three different saxophones, including the forgotten Cmelody saxophone. The use of improvisation and the emphasis on sax gives the proceedings a distinctly jazzy flavor, even though they play some very familiar Italian favorites, like “Tu Vuo ‘Fa’ L’Americano” (or, in English, “You Gotta Be Americano”). Throughout this very entertaining CD, I was reminded of the accordion-reed lead of the great Joe Mooney quartet.

While Quadro Nuevo is light and pleasant, pianist Enrico Pieranunzi’s music will satisfy the more hard core American jazz fan. In 1981, producer Hal Willner masterminded a brilliant album called “Amarcord Nino Rota” (featuring, in addition to many avant-garde players, Branford and Wynton Marsalis), and it’s only fitting that a leading Italian jazz pianist should pay homage to Rota and the other composers who wrote for Fellini. Here, the remaining members of Mr. Pieranunzi’s quintet are Canadian-British trumpeter Kenny Wheeler, and three Americans: saxophonist Chris Potter, bassist Charlie Haden, and drummer Paul Motian.

All the solos are highly evocative, suggesting a dreamy, “Amarcord”-like quality, as if the players were collectively reminiscing about their youth. Even, Mr. Potter, who was only 32 at the time of the sessions, sounds warmly introspective throughout, especially playing soprano on “Le Notte di Cabiria.” Charlie Haden, who has dedicated the most recent part of his career to celebrating great melody, is the perfect bassist for such a project, and he makes these familiar themes sing out as much as any of the horns. “La Strada,” a Rota tune that became well known over here (it was sung beautifully by Perry Como), is surprisingly done somewhat faster than usual. It is still achingly nostalgic, though, as a vehicle for Mr. Potter’s tenor.

Six different ensembles are spotlighted in the JazzItaliano festival this week, and their memberships constitute a variety of Italians who still are based in their native land, Italian-born musicians who currently reside in New York, and, in some cases, Italian-Americans. Two guitarists will be featured: Fabruzui Scotti plays one night (Wednesday) with his trio at Le Jazz Au Bar and Andrea Braido with Pippo Matino (bass) for two nights, tonight and tomorrow, at the Jazz Gallery. Two other ensembles are brass centric: trumpeter Fabio Morgera with an Italian rhythm section and two firstrate local players, George Garzone (tenor) and Pete Malinverni (piano), and “Le Tromba del Re,” with a front line of Fabrizio Bosso and Flavio Boltro on trumpets and flugelhorns.

The two groups I’m most looking forward to hearing are those of Rosario Giulani at Smoke (March 4 to 6) and Roberto Gatto at Sweet Rhythm (March 1 to 4). Mr. Gatto is an outstanding drummer-bandleader in the tradition of Art Blakey, Roy Haynes, and Max Roach. His band features sax and trombone in the front-line. Senor Giulani is, like Senor di Battista, still another one of those incredibly fluent saxophone virtuosos who, thanks to jazz education, seem to be proliferating on both sides of the Atlantic – and I can never get enough of those.

“Jazzitaliano in New York” continues until March 6. For more information, visit www.jazzitaliano.com.