Next Big Future has produced a number of articles recently on breakthrough propulsion. Here is a summary of the current state of the art on these speculative systems.
Low-Thrust Propulsion systems are too weak to be used to launch from the surface of a planet, but have extreme efficiency and high exhaust velocities. Basically, a space probe like Dawn had the propulsive force equivalent two sheets of paper held on your hand. Yet it could thrust for years with this system. Over the course of eight years, it thrust over 2000 days, with a change in velocity of 24,000 mph/ 38,000 kph. That's more than it took to launch it into interplanetary space to begin with, and while using less than 937 pounds/425 kg of propellant on board. Ion drive takes a slowly-released gas, gives it an electrostatic charge, and then accelerates it in a magnetic field and neutralizes the charge as it exits the engine. Newer systems, such as the speculative EM-Drive, promise to work with no on-board propellant.
While creating full-antimatter is very, very difficult and energy-intensive, making simple positrons (antimatter electrons) is fairly simple. A group is working on a system that can fit on a cubesat for a demonstrator. Rather than make full anti-matter atoms and trying to store them and use them later in an engine, a positon system is basically an ion drive system like Dawn uses, but using positrons made on the ship itself instead of ionized gas. Positrons are naturally produced by radioactive materials. So a sample of a radioisotope is placed on board, and the positrons it emits are cooled enough to allow them to be directed and accelerated in a magnetic field like in a conventional ion drive. The resulting particles exit the vehicle at ten percent the speed of light. If idealized, these systems could get to Mars in weeks, Pluto in months, and Alpha Centauri in 40 years. The developer, Positron Dynamics, plans to launch a cubesat demonstrator somewhere between the middle of this year and next year. The near term application of these motors would be as a more-efficient substitute for ion drive in positioning communications satellites. In the longer term, they would be very useful for asteroid mining.
To make a complex system a bit easier to understand, a small fuel pellet is compressed by a magnetic field and bombarded with particles. It undergoes nuclear fission and the plasma is further compressed by the magnetic field. The combination of compression and the heat, self-generated plasma magnetic field, and particle mix of the tiny fission explosion can trigger a fusion explosion, similar to the way an atomic bomb can trigger a thermonuclear (fusion) bomb. As with the Positron example, this could theoretically get a vessel to Mars in a month. It would generate exhaust ISP(the efficiency of a rocket) of 30,000 seconds. By contrast, a typical chemical rocket has an ISP under 450 seconds.
This is similar to EM drive, in that it depends on A) very tiny effects that can barely be observed and B) processes that defy conventional physics, but appear to be explained by some aspects of science. When I say barely observable, we are talking 2 to 12 micronewtons. A tiny bottle-rocket has 3 newtons of average thrust, or a million times as much as these experimental engines. That said, they hope to get up to 10-20 millinewtons in a decade or two. That said, a microthrust, propellantless system has no theoretical output limit, because there is no propellant to be used up. This is why those who propose systems like this often speak of starships.
The positron system seems quite logical, and can be demonstrated soon. It has limits, though, because it relies on radioactive decay of highly radioactive materials. This can be scaled down into tiny systems. The PuFF system will require a lot of expensive research to get worked out, over a fairly long time period. That said, it has a lot of similarities to Project Daedalus, and could be used for a simple starship probe eventually. Such a system is at least a decade away, and would be quite large. Mach Effect is a lot like EM Drive, but while the effects are more explainable, they seem to be even less powerful. To power a vehicle with any velocity, it also has to be scaled up quite a bit.
With new systems that can provide fission and possibly fusion electrical power coming online very shortly, systems that require lots of electricity (advanced ion drive, these propellantless systems, and so on) may be arriving sooner rather than later. Expect to see something interesting in this range within five to ten years.