Just to add to Drakkith's reply- 'splitting' a single element into a doublet increases the degrees of freedom for the designer from 3 (two surfaces plus element thickness) to 7 (4 surfaces plus three spacings). Not all aberrations can be corrected in this way: chromatic aberrations require the use of different materials to correct the dispersion. Aspherical elements can be used to reduce the number of surfaces, and the location of the aperture stop provides an additional degree of freedom. The aberrations also depend on object distance: a lens may be well corrected for viewing distant objects, but horrible for viewing nearby objects. Similarly, correcting the aberrations over the entire field of view becomes more difficult as the field of view increases.
Even so, not all monochromatic aberrations can be fully corrected- the ideal design may require a zero-thickness or even negative thickness. Buchdal's and Kingslake's books are excellent resources. For example, Buchdal presents a detailed solution for a Cooke triplet: 6 surfaces and 5 spacings result in 5 dense pages of residual aberrations- only the primary and secondary monochromatic aberrations. This is why modern lens design is performed on a computer.