Tempo rubato (Italian for: stolen time) is a musical term referring to expressive and rhythmic freedom by a slight speeding up and then slowing down of the tempo of a piece at the discretion of the soloist or the conductor. Rubato is an expressive shaping of music that is a part of phrasing.
Types of rubato 
[...] this loose term "rubato." It describes the practice of playing with expressive and rhythmic freedom. Specifically "tempo rubato" [...] some time is "robbed" from one passage or group of notes and given to another.
One can distinguish two types of rubato: in one the tempo of the melody is flexible, while the accompaniment was kept in typical regular pulse (yet not rigidly in mechanical fashion; but adjusting to the melody as necessary—see below). Another type affects melody and accompaniment. While it is often associated with music of the Romantic Period, classical performers frequently use rubato for emotional expressiveness in all kinds of works.
Rubato, even when not notated, is often used liberally by musicians; e.g. singers frequently use it intuitively to let the tempo of the melody expressively shift slightly and freely above that of the accompaniment. This intuitive shifting leads to rubato's main effect: to make music sound expressive and natural. Frédéric Chopin is often mentioned in context with rubato (see Chopin and Rubato).
While rubato is often loosely taken to mean playing with expressive and rhythmic freedom; it was traditionally used specifically in the context of expression by speeding up and then slowing down. In the past expressive and free playing (beyond only rubato) was often associated with the terms "ad libitum":
Tempo rubato (or a tempo rubato) means literally in robbed time, i.e., duration taken from one measure or beat and given to another, but in modern practice the term is quite generally applied to any irregularity of rhythm or tempo not definitely indicated in the score.
The terms ad libitum, (ad lib.), a piacere, and a capriccio, also indicate a modification of the tempo at the will of the performer. Ad libitum means at liberty; a piacere, at pleasure; and a capriccio, at the caprice (of the performer).—Music notation and terminology (1921) by Karl Wilson Gehrkens
A tempo rubato. Lit. "in robbed time", i. e. time in which, while every bar is of its proper time value, one portion of it may be played faster or slower at the expense of the remaining portion, so that, if the first half be somewhat slackened, the second half is somewhat quickened, and vice versa. With indifferent performers, this indication is too often confounded with some expression signifying ad libitum.—A dictionary of foreign musical terms and handbook of orchestral instruments (1907) by Tom S. Wotton
The opinion given by Tom S. Wotton, that "every bar has its proper time value" may be regarded as an inaccurate description: Karl Wilson Gehrkens mentions "duration taken from one measure [...] and given to another" which implies bars of differing duration. Rubato relates to phrasing; and since phrases often go over multiple bars; it is often impossible (and also not desired) for each bar to be identically long.
There is no absolute rhythm. In the course of the dramatic developments of a musical composition, the initial themes change their character, consequently rhythm changes also, and, in conformity with that character, it has to be energetic or languishing, crisp or elastic, steady or capricious.
[...] Rubato must emerge spontaneously from the music, it can't be calculated but must be totally free. It's not even something you can teach: each performer must feel it on the basis of his or her own sensitivity. There's no magic formula: to assume otherwise would be ridiculous.
Performers also frequently show a tendency to speed up and slow down when this is not indicated in the score. Such modifications of tempo typically occur in relation to phrase structure, as a way of marking phrase boundaries.
Tempo Rubato is a potent factor in musical oratory, and every interpreter should be able to use it skillfully and judiciously, as it emphasizes the expression, introduces variety, infuses life into mechanical execution. It softens the sharpness of lines, blunts the structural angles without ruining them, because its action is not destructive: it intensifies, subtilizes, idealizes the rhythm. As stated above, it converts energy into languor, crispness into elasticity, steadiness into capriciousness. It gives music, already possessed of the metric and rhythmic accents, a third accent, emotional, individual, that which Mathis Lussy, in his excellent book on musical expression, calls l'accent pathètique.
Variations of Tempo, the ritardando, accelerando, and tempo rubato, are all legitimate aids demanded by Expression. [...] use is determined by sound judgment and correct musicianly taste.
many [...] hold very strange ideas about [rubato]. There are those who suffer with a technique too unreliable to keep them steady in prolonged rapid passages and when they run away with the tempo they palm their weakness off as "tempo rubato". Then there are those who believe that the use of tempo rubato begins with the works of Chopin and that it must not be employed in any music written before him. There are also the arch-pedants who insist that tempo rubato not only begins but also ends with Chopin's compositions. All of which is, of course, pure cant, the bulwark of ignorance and bigotry.
Because the purpose of rubato is to add a sense of improvisatory freedom to the performance, one should avoid using the same kind of rubato repeatedly in a piece. Stretching or rushing successive phrases in the same way creates a monotonous sense of predictability that defies the purpose.
"Accompaniment yields/adjusts to melody" 
Dictionary definitions of musical concepts (such as rubato), may sometimes cause misinterpretations – often because of explanations that disregard artistic musical expression, or fail to recognize artistic musical expression as the actual source of the concept discussed. Concerning the type of rubato, in which the accompaniment is kept regular, it must therefore be mentioned that this does not refer to absolute strict regularity: the accompaniment does still give full regard to the melody (often singer or soloist) and yields tempo where necessary. It is critically viewed, when the accompaniment is just played absolutely regularly and with complete disregard for the melody:
It is amusing to note that even some serious persons express the idea that in tempo rubato "the right hand may use a certain freedom while the left hand must keep strict time." (See Frederick Niecks' Life of Chopin, II, p. 101.) A nice sort of music would result from such playing ! Something like the singing of a good vocalist accompanied by a poor blockhead who hammers away in strict time without yielding to the singer who, in sheer despair, must renounce all artistic expression.
[...] nothing in general can be more disagreeable than this species of brilliant accompaniment, where the voice is only considered as an accessory and where the accompanier, without regarding the taste, feeling, compass, or style of the singer, the pathos of the air, or sense of the words, either mechanically runs though the prescribed solemnity of the adagio, with the one two three precision of the metronome, or rattles away without mercy through the allegro whenever an occasion presents itself for the luxuriant ad libitum introduction of turns, variations, and embellishments.—Almack's revisited: or, Herbert Milton (1828) by Charles White)
Some people, evidently led by laudable principle of equity, while insisting on the fact of stolen time, pretend that what is stolen ought to be restored. [...] The value of notes diminished in one period through accelerando, cannot always be restored in another by ritardando. What is lost is lost.
[...] a Metronome is apt to kill the finer Time-sense implied by Rubato.
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