The lowest known is 17 givens in general Sudoku, or 18 when the positions of the givens are constrained to be half-turn rotationally symmetric. It is conjectured that these are the best possible, evidence for which stems from extensive randomised searching:
Gordon Royle has compiled a list of (currently) 36628 17-clue puzzles, each of which is unique up to isomorphism. None of these was isomorphic to a symmetric puzzle, nor contained a 16-clue puzzle.
An independent construction of 700 distinct 17-clue puzzles found only 33 not already on an earlier, size 32930, version of Royle's list. From that, the MLE estimate of the number of 17-clue grids was c. 34550. If the construction methods were truly random and independent then this would imply a negligible probability of there being a 16-clue puzzle waiting to be discovered -- since any such "16" would give rise to 65 17-clue puzzles, all of which were somehow missed in the search.
The most fruitful set of clue positions, in terms of number of distinct 17-clue puzzles they admit, from Royle's list have been exhaustively searched for 17-clue puzzles. All 36 puzzles found by this process were already in Royle's list. All 34 puzzles on the next most fruitful set of clue positions were also in Royle's list.
The most fruitful solution grids (in the same sense) have been exhaustively searched for 16-clue puzzles using CHECKER with no success. This includes one, "strangely familiar", grid that yields at least 29 different 17-clue puzzles.