Current manufacturer initiatives
As to the current state of fully autonomous passenger jets technology (in 2017), there is some interest by manufacturers in the technology, particularly by Boeing, but it would be a stretch to say that manufacturers are working hard to get fully autonomous passenger jets commercially ready soon.
At the Paris Airshow in 2017, Mark Sinnett, vice president at Boeing responsible for innovative future technologies, clarified that, as paraphrased by the Wasington Post, "The artificial intelligence that Boeing will test will be capable of making decisions normally made by pilots. Sinnett said that the technology could be used to reduce the number of pilots required for long-haul flights or, in some situations, allow fully autonomous flying"
Earlier in 2017 Sinnett had stated that “We are not smart enough to preprogram all those things. The machine has to be capable of making the same set of decisions [as a pilot],” Sinnett said. “If it can’t, we cannot go there.”
In a talk at the American Institute of Aeronautics and Astronautics SciTech Conference in January 2015, John Tracy, Boeing’s chief technology officer, said: “Some of our freighter customers are asking us for those [autonomous airplane] systems today.”
He went on: “We are quite confident that technologically, the toolkit is filled. With respect to a commercial airplane, there is no doubt in our minds that we can solve the problem of autonomous flight. It’s a question of certification
procedures, regulatory requirements and, even more significantly, public perception. "
A spokesperson for Airbus said in 2016: “Airbus is not developing an autonomous airplane.”
And a spokesperson for the FAA, which would be responsible for certifying an autonomous plane, said in 2016 that: “The FAA has no current unmanned aircraft certification projects in the transport category, nor has anyone engaged the agency on such a project.”
Chinese company Ehang has produced a passenger drone, the Ehang 184, that can fly a 25 minute preprogrammed route from take-off to landing. This doesn't preclude many of the issues described below. The technology is still in test flights in 2017.
While previous advances in autopilot and even crashes due to misuse of autopilot have not caused widespread distrust, John Tracy of Boeing stated that public perception is very important to successful implementation. My opinion is that when things start to go wrong in the cockpit, many people would trust a human with thousands of hours of experience over a computer with millions of hours of testing. The lack of faith in self-driving car technology seems to confirm this tendency.
What decisions currently require a pilot?
Preflighting the airplane: The pilot needs to review the entire plane before flying. This includes reviewing the list of items that need non-urgent repairs and deciding if he's comfortable with the safety margin on the plane. Sometimes this relies on predicting whether the plane will encounter certain weather conditions.
Weather: A pilot has to make the decisions on whether or not a route is safe to fly or if conditions will get worse. In-flight, this may require a decision to declare an emergency. A pilot with detailed radar-based weather maps may even be able to choose a route right through a gap between dangerous storm cells. A computer would have to be better than a pilot at predicting how safe a particular route is, in order to avoid situations like Southwest Airlines Flight 1248 or many other crashes where weather was a factor.
Flying visual approaches: Even though autoland technology has existed since 1964, it's still uncommon to have an airport equipped with all runways enabled for zero-visibility approaches. (see this question). Less than 1% of all landings are currently done using autoland.
Taxiing: Taxiing is currently done manually, and by visual reference. Detailed GPS maps and indications of where you are on the taxiways exist but the plane can't drive through them automatically. Automatic taxiing by use of external towing vehicles is an area of active research (see here or here)
Communication with ATC: Airspace clearances, weather information, taxiway directions, traffic diversion, declaring an emergency and more are all handled verbally with ATC. Partial replacements for much of this has been made with ACARS, TCAS, and advanced radios, but there remains a lot of work to be done in taking ATC verbal communication completely out of some planes.
Avoiding VFR traffic and terrain: Not all traffic is transponder-equipped and not all terrain is recorded in an accurate database or covered by an enhanced ground-proximity awareness device, so we rely on pilots to avoid these. Inaccurate charts are rare in modern times so the risk here is probably acceptable, but imperfections in maps have contributed partially to crashes like Lockheed L-1329 JetStar N400M or World Airways Flight 802
Identifying when things are going wrong: Although some malfunctions, like total engine failure, are easy to identify, other issues are trickier to sort out and require visual confirmation or pilot judgement. For example, if two flight instruments disagree, which one is malfunctioning?
Making a plan of action for equipment malfunctions: Can a plane still fly to an airport with an equipment problem or do you need to declare an emergency? Is an unusual flying technique or reduced flight envelope required after the failure? Not to be an alarmist, but exploding engines, landing gear failures, equipment fires, and tire explosions can happen several times in the industry in a single year, not to mention much more common issues like indicator lights not working, bird strikes, compressor stalls, engine malfunctions, and autopilot disengagement.
Route planning: Pilots still often manually read approach charts and sectional charts and enter the resulting route into the FMS.
Running some of the manual troubleshooting: Current hardware doesn't allow the flight control computers to pull breakers, disengage generators, or restart engines. Currently the philospophy that I've seen in autopilots is "if something unusual is going on, disengage and pilots will take care of it".
Turbulence and windshear: Autopilots can currently handle these tough wind conditions, but pilots hand-flying the aircraft are usually better and provide a larger margin of safety.
On-board medical problems and other passenger problems: Someone needs to make a judgement call as to whether an onboard medical problem or someone getting violent over yoga would require a diversion, and currently this decision is up to the pilot and other flight crew.
Can't we handle some of this remotely?
You may ask why we can't simply remotely handle some decisions from mundane weather rerouting to emergency procedures, but that only eliminate some of the hurdles. Currently the technology for remote piloting is not mature enough (see these questions about engineers on board, converting planes to drones, and uploading FDR data). I've seen some of the latest generation connectivity technology for aircraft and even in 2017 it still has reliability, latency, cost, or speed issues that make it unsuitable for handling a matter of life or death, particularly in area like oceans where satellites are your only option.
The topic of advances towards fully autonomous aircraft is a broad and complex topic. Here are some articles for further reading: