  We give a modern computational introduction to the S,K combinators invented
by Moses Sch\"onfinkel in 1920, and present a variety of new results and ideas
about combinators. We explore the spectrum of behavior obtained with small
combinator expressions, showing a variety of approaches to analysis and
visualization. We discuss the implications of evaluation strategies, and of
multiway systems representing all possible strategies. We show how causal
graphs introduced in recent models of fundamental physics can be applied to
combinators, as well as describing how combinators introduce a new form of
treelike separation. We give a variety of new results on minimal combinator
expressions, as well as showing how empirical computation theory and
computational complexity theory can be done with combinators. We also suggest
that when viewed in terms of ongoing computation, the S combinator alone may be
capable of universal computation.
