The Hippocratic Corpus is composed of around sixty early works. Émile Littré spent twenty-two years (1839–1861) working diligently on a complete Greek edition with French translation.
Beginning in 1967, Jacques Jouanna produced a modern critical edition of the Greek text with a French translation and commentary in the Collection Budé.
The first English translation appeared in the Loeb series in 1923–1931 and 1988 and 2012.
For a good introduction to the corpus see:
Smith, W. D. (1979). The Hippocratic Tradition. Ithaca and London: Cornell University Press.