The subject of real algebraic plane curves began in 1876, when Harnack proved
that a smooth real plane curve of degree d can have at most
1 + (d-1)(d-2)/2 ovals, and then he constructed plane curves of
any degree achieving this bound.
Harnack's inductive construction is very elegant and straightforward.
A sequence of web pages
illustrating Harnack's construction of his sextic.
Much work was done on finding prohibitions as well as new constructions in the
ensuing 100 years.
A breakthrough with important consequences for other fields was achieved by
Viro around 1980 with his construction of T-curves based upon what are
now called toric deformations.
A sequence of web pages illustrating Viro's construction of the Harnack
sextic.
A sequence of web pages
illustrating Harnack's construction of his sextic.