In 1934, Jordan, von Neumann and Wigner gave a nice classification of finite-dimensional simple Jordan algebras that are 'formally real', meaning that a sum of squares is zero only if each term in the sum is zero. In 1983 Zelmanov generalized this to all simple Jordan algebras, but unlike the original result, Zelmanov's 'classification' is not a neat list. What about formally real Jordan algebras, not necessarily finite-dimensional? What does the classification of these look like?
The formally real Jordan algebras include the class of JB-algebras, a class of normed Jordan algebra which is the Jordan algebra equivalent of C*-algebras. From this you might imagine that obtaining a complete classification is a rather non-trivial task. In this class you also find JW-algebras, which are JB-algebras with a predual, thus corresponding to von Neumann algebras. Some of the concepts from C*- and von Neumann algebra theory carry over to the Jordan algebra setting, but this margin is too narrow to summarize what is known. With apologies for tooting my own horn here, in 1984 I coauthored a book “Jordan Operator Algebras” with Erling Størmer which pretty much summarized the state of the art at the time. (Since then I have left that field, so I don't know if a lot has happened since.)