You may have known from the movie Amadeus that in the old time, composers usually took up the responsibility to conduct their own works in concerts. This is certainly not a bad idea. After all, who knows the works better than the composers themselves? Also, as the ensemble was usually under-rehearsed in the premiere (even for performances nowadays...?) struggling with possibly unfinished hand-written score, it was better to let composers handling their own business...
'Mozart conducting Mozart' (Source: Amadeus, 1984)
The tradition of composer being a conductor can be traced back to Mendelssohn, Berlioz, Wagner. Thanks to modern technology, we are fortunate to hear some of the foremost composers showing the most authentic rendition of their pieces. Here is an example:
Richard Strauss conducting Don Juan
(who knows what the musicians were thinking when they first played this notoriously difficult tone poem...)
Mahler was the subscription conductor of the Vienna Philharmonic during the years 1898-1901. Although there is no available recording of his conducting, we can have a glimpse of his own interpretation and musical insight from his piano playing:
Mahler playing excerpts from Symphonies No.4 and No. 5
Finally, legendary conductor of the Berliner Philharmoniker, Wilhelm Furtwängler, reversely had a lesser-known identity as a composer. Here is the recording of his complicated and lengthy Symphony No. 2:
Furtwängler conducting Symphony No. 2



