Cadenza often refers to a portion of a concerto in which the orchestra stops playing, leaving the soloist to play alone in free time (without a strict, regular pulse) and can be written or improvised, depending on what the composer specifies. This normally occurs near the end of the first movement, though it can be at any point in a concerto; an example is Tchaikovsky's First Piano Concerto, where in the first five minutes a cadenza is used. It usually is the most elaborate and virtuosic part that the solo instrument plays during the whole piece. At the end of the cadenza, the orchestra re-enters, and generally finishes off the movement on their own, or, less often, with the solo instrument.
The cadenza was originally, and remains, a vocal flourish improvised by a performer to elaborate a cadence in an aria. It was later used in instrumental music, and soon became a standard part of the concerto. Originally, it was improvised in this context as well, but during the 19th century, composers began to write cadenzas out in full. Third parties also wrote cadenzas for works in which it was intended by the composer to be improvised, so the soloist could have a well formed solo that they could practice in advance. Some of these have become so widely played and sung that they are effectively part of the standard repertoire, as is the case with Joseph Joachim's cadenza for Johannes Brahms' Violin Concerto, Beethoven's set of cadenzas for Mozart's Piano Concerto no. 20, and Estelle Liebling's edition of cadenzas for operas such as Donizetti's's La fille du Régiment and Lucia di Lammermoor.
Nowadays, very few performers improvise their cadenzas, and very few composers have written concertos or vocal pieces within the last hundred years that include the possibility of an improvised cadenza.
Perhaps the most notable deviations from this tendency towards written (or absent) cadenzas are to be found in jazz, most often at the end of a ballad, though cadenzas in this genre are usually brief and somewhat immaterial. Saxophonist John Coltrane, however, usually improvised an extended, spell-binding cadenza when performing "I Want To Talk About You", in which he showcased his predilections for scalar improvisation and multiphonics; the recorded examples (see "Coltrane Live At Birdland" and "Afro Blue Impressions"-- both live recordings) of Coltrane's "I Want To Talk About You" are approximately 8-minutes in length, with Coltrane's unaccompanied cadenza taking up approximately 3-minutes. More sardonically, Jazz critic Martin Williams once described Coltrane's improvisations on "Africa/Brass" as "essentially extended cadenzas to pieces that never get played." Equally noteworthy is saxophonist Sonny Rollins' shorter improvised cadenza at the close of "Three Little Words" (from his album "Sonny Rollins on Impulse!").
Cadenzas are also found in instrumental solos with piano or other accompaniment, where they are placed near the beginning or near the end or sometimes in both places. (e.g. "The Maid of the Mist," cornet solo by Herbert L. Clarke, or a more modern example: the end of "Think of Me" where Christine Daaè sings a short but involved cadenza, in Andrew Lloyd Webber's The Phantom of the Opera.)