  This paper is an elaborated version of the material presented by the author
in a three hour minicourse at "V International Course of Mathematical Analysis
in Andalusia," Almeria, Spain, September 12-16, 2011. Part I is devoted to an
exposition of the properties of derivations on various algebras and triple
systems in finite and infinite dimensions, the primary questions addressed
being whether the derivation is automatically continuous and to what extent it
is an inner derivation. Part II discusses cohomology theory of algebras and
triple systems, in both finite and infinite dimensions. Although the cohomology
of associative and Lie algebras is substantially developed, in both finite and
infinite dimensions, the same could not be said for Jordan algebras. Moreover,
the cohomology of triple systems has a rather sparse literature which is
essentially non-existent in infinite dimensions. Thus, one of the goals of this
paper is to encourage the study of continuous cohomology of some Banach triple
systems. Part III discusses three topics, two very recent, which involve the
interplay between Jordan theory and operator space theory (quantum functional
analysis). The first one, a joint work of the author, discusses the structure
theory of contractively complemented Hilbertian operator spaces, and is
instrumental to the third topic, which is concerned with some recent work on
enveloping TROs and K-theory for JB*-triples. The second topic presents some
very recent joint work by the author concerning quantum operator algebras. One
section in Part III is devoted to the subject of contractive projections, which
play an important role in the structure theory of Jordan triples.
