New looks at classic scores by Russian composers highlight dance in New York City and its environs over the summer. London’s Michael Clark brings his “Stravinsky Project” to the Rose Theater June 4-7 as part of Lincoln Center’s Great Performers “New Visions” series. Get ready to see how the renegade Mr. Clark reimagines Stravinsky’s “Les Noces,” “Apollo” and “Le Sacre du Printemps,” each of which was first choreographed early in the 20th century. Mr. Clark’s company will perform to live music.

At Bard College on July 4, Mark Morris will present the world premiere of his “Romeo and Juliet, on Motifs of Shakespeare” as part of Bard’s annual SummerScape Festival, which focuses this year on “Prokofiev and His World.” No work of Prokofiev’s is more widely performed than his ballet “Romeo and Juliet,” first choreographed in 1940 by Leonid Lavrovsky for the Kirov Ballet in then-Leningrad. But this won’t entirely sound like what we’re used to: Mr. Morris is working with a new version of the score prepared by Princeton University musicologist Simon Morrison, culled from documents in a Moscow archive. According to Mr. Morrison, this represents the first time Prokofiev’s ballet will be performed as the composer originally intended, restoring elements of the score and the scenario that did not pass muster with the tightly controlled world of Soviet aesthetics under Stalin. Mr. Morris’s work will be performed by his Mark Morris Dance Group, and Prokofiev’s score will be conducted by Leon Botstein, leading the American Symphony Orchestra.

Throughout June, America’s two leading ballet companies face-off, as they do annually, at Lincoln Center. American Ballet Theatre’s Metropolitan Opera season goes through July 12, while New York City Ballet’s annual spring season runs at the New York State Theater through June 29. NYCB’s Jerome Robbins Celebration dominates its performance calendar. On June 18, NYCB performs a one-time-only program to commemorate principal dancer Damian Woetzel’s final performance with the company. Mr. Woetzel joined NYCB in 1985, and has continued to mature artistically right through his most recent performances this season.

Dance makes its presence felt at the annual Lincoln Center Festival, with five performances of William Forsythe’s 1988 work, “Impressing the Czar,” by the Royal Ballet of Flanders, July 17-20. The choreographer made it for his Ballett Frankfurt but, until the RBF revived it in 2006, the piece had not been performed since 1995. New York audiences will be familiar with its central section, “In the Middle, Somewhat Elevated,” but not with the rest of Mr. Forsythe’s far-ranging and freewheeling treatise on dance, history, and civilization.

The Joyce Theater has rounded up contemporary dance companies from all over the world for the summer months. “The French Collection” encompasses one-week seasons by Compagnie Heddy Maalem, making its American debut, Compagnie Maguy Marin, and Ballet Biarritz. Each troupe has something definite to say: For Maalem, it’s a cross-referencing of African dance and Stravinsky, while Marin and Biarritz are doing pieces inspired by Samuel Beckett and Velasquez, respectively.

Pilobolus needs no introduction as it returns to the Joyce for its annual four-week season. Three new pieces are going to be presented, including a collaboration with acclaimed puppeteer Basil Twist, whose version of “Petruchka” attracted a lot of attention earlier this spring.

Robert Battle’s Battleworks Dance Company shares a week in July with Keigwin + Company, while Hubbard Street Dance Chicago will show five New York premieres during its two-week season.

Finally, Lincoln Center Out of Doors brings dance to the band shell in Walter Damrosch Park in August. There will be the premiere of “Summer of Love,” Armitage Gone’s collaboration with Burkina Electric, and Doug Elkins’s “Fräulein Maria,” a highly personal and comic tribute to “The Sound of Music” — something of a Manhattan favorite since it was first performed in 2006.