There is no universal agreement on this topic. While there may be a general consensus, you can always find a scholar arguing the opposite view.
The standard ornaments in the practice of Mozart and Haydn can be derived from the earlier, larger Baroque repertoire. Leopold Mozart and C. P. E. Bach are standard sources here. The general consensus today is to start all ornaments on the beat, except perhaps the acciaccatura which is so short that distinguishing is difficult. More often than not this is carried forward into Beethoven as well. For example, Arrau's edition recommends this practice. Broadly speaking, the generation before him is most likely to be heard beginning ornaments before the beat. Czerny Op. 500 is a useful source here.
The interpretation of the acciaccatura is fairly consistent through time, the appoggiatura less so. The length it absorbs from the following note in the Romantic occurrences that spring to mind is very much at the performer's discretion. On the other hand, its notation in Mozart and Haydn is more grammatical. Finding a small quaver appoggiatura preceding a crotchet, typically one would simply play two quavers, likely with a small extra "leaning" accent.
The 19th century saw the appearance of a large number of primers with something to say about this topic. The general view was that appoggiaturas begin on the beat. It is of course hard to tell how closely these texts reflect the contemporary performance practice.
In response to a comment that an acciaccatura is distinguished by a stroke through the note: this is the general view but the picture is more complex than that. The forceful keyboard acciaccatura and the short appoggiatura in music for various instruments have both been notated with the stroke, in the latter case to distinguish it from the long appoggiatura.