How much is autopilot on commercial flights today?

How much of an average commercial flight on a large airline and
aircraft today is done automatically, via autopilot, autoland, and so
on? Including takeoff and landing. I know that the bulk of a flight
is on autopilot, which makes sense since autopilots are good at flying
planes and that's lots of hours to sit holding a yoke, but what about
the fun parts, such as takeoff and landing? Systems exist to fully
automate both, but are such systems routinely installed and used
today? Do airlines have policies that require or prohibit the use of
such systems under normal conditions? I've heard that in cruise
flight, at least, airlines want their pilots to stay on autopilot as
much as possible, to save fuel.

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