Yes to take-off, but no to landing...
Well, it depends on the type of auto-pilot of course as some are more advanced than others, so the answer's based on the 'best' ones in use atm.
Having just seen the answer above mine (written at much the same time), i've done a quick search & the answer's yes to both -
Modern autopilots generally divide a flight into taxi, take-off, ascent, level, descent, approach, landing, and taxi phases. Autopilots exist that automate all of these flight phases except the taxiing.