Short answer: Yes.
Longer answer: Yes, but there are models that are specifically made to fit into plane seats and clip into the seat using the seat belt and sometimes a seat belt extender. For this reason the seats must have an independent harness to hold the child (as most seats do) and can not be used in "booster" configuration as this usually requires a shoulder belt and doesn't have the independent harness.
Your car seat must show that it has met the approval standards for use on an airplane, this must be indicated by a tag/label that says:
This restraint is certified for use in motor vehicles and aircraft,
which means that the model has undergone testing to meet the required standard. The FAA has standards for this, and airlines also accept similar standards from places outside the USA.
You can find more information about the standards, what sort of conditions apply and similar details in this advisory PDF from the FAA.
Some salient points about approved devices from linked document:
Section 10.C: CRS Restrictions. No aircraft operator may permit a child to occupy a booster-type,
vest-type, harness-type, or lap-held CRS during takeoff, landing, and movement on the surface,
except when the CRS has been approved by the FAA through a TC, STC, TSO, under
§ 21.305(d) (2010 ed.), or under § 21.8(d). Booster-type, vest-type, and harness-type CRSs
approved by the FAA through a TC, STC, TSO, under § 21.305(d) (2010 ed.), or under
§ 21.8(d), may be used during all phases of flight.
Section 10.F.1: A CRS with a base that is too wide to fit properly in a seat with rigid armrests can be
moved to a seat with moveable armrests that can be raised to accommodate the CRS in the same
class of service.
Section 11: 11. SEAT DIMENSION DISCLOSURE. Consistent with the FAA Modernization and Reform
Act of 2012, § 121.311(k) requires air carriers conducting part 121 operations to make available
on their Web sites the width of the narrowest and the widest passenger seats in each class of
service for each airplane used in passenger-carrying operations. This rule facilitates the use of a
CRS onboard an airplane and provides greater information to assist a caregiver to determine
whether a particular CRS will fit in an airplane seat.
Section 13 (summary of points):
• The CRS should have a solid back and seat,
• The CRS should have internal restraint straps installed to securely hold the child in the
CRS, and
• The CRS must have a label showing approval for aviation use.
Personally I've flown on many international and domestic flights with children in a child seat from Diono (IIRC, one of the Radian models) and never had a problem with use on the plane in any configurations (forward/rear facing) nor any problems checking in, at the gate, etc. The Diono one was also folding which meant much less hassle getting it onto the plane and while moving/waiting around the departure/arrival areas. Good for giving them a familiar environment to nap in while waiting around too. The main downside was that at a certain age the seat enables them to reach the screen, so be prepared for lots of turning that on and off on-board.