Spherical, coma, astigmatism, field curvature and distortion are main aberrations and they present in a lens. If a lens is used to image an object these aberrations will affect the image of the object. If we try to reduce the 3rd order aberrations, next, higher order aberrations will come into picture.
The Seidel aberrations (Spherical, Coma, Astigmatism, etc.) really represent a Taylor series (polynomial) expansion -- series like this must be rapidly converging in order to have any useful meaning. One of the problems that arises in aberration correction, especially in high NA systems, is that the series doesn't converge as well and the high order terms become non-neglible and sometimes dominant. Historically (prior to computer aided design and rapid real ray tracing) the most reliable way to make sure this doesn't happen is to keep the offense against the sine condition (OSC) small, assuring that the third order aberrations are dominant. Now, one can optimize using real ray tracing in such a way that all orders are included, so the problem becomes one of making sure that the maximum slope of any given surface is not too large. (Large slopes are hard to fabricate and very difficult to measure.)
For more information on lens design, I highly recommend Prof. Julie Bentley's courses at the University of Rochester (and offered in short form at several SPIE conferences and as a UR Summer short course). UofA of course also has excellent offerings as do other Universities.
Deformable mirrors or PAL-LCdd devices can certainly do that with some adaptive optics arrangement actively. If it is just a constant vaue to be removed, known value not changing can even work with a phase plate, there is plenty of literature over there on that. Design does also work if this is your case, as Thomas commented by trying to reduce the OSC.
However, take care on whether you really need reducing them or removing them. Nothing is free and you will spend your degrees of freedom or your money and time. In general, the larger order yo go in the Taylor series the more prone you are to unstabilize your compensation algorithm, because of noise effects, fitting errors or measurement accuracies.
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