  I analyze the hierarchy of large cardinals between a supercompact cardinal
and an almost-huge cardinal. Many of these cardinals are defined by modifying
the definition of a high-jump cardinal. A high-jump cardinal is defined as the
critical point of an elementary embedding $j: V \to M$ such that $M$ is closed
under sequences of length $\sup\set{j(f)(\kappa) \st f: \kappa \to \kappa}$.
Some of the other cardinals analyzed include the super-high-jump cardinals,
almost-high-jump cardinals, Shelah-for-supercompactness cardinals,
Woodin-for-supercompactness cardinals, \Vopenka\ cardinals, hypercompact
cardinals, and enhanced supercompact cardinals. I organize these cardinals in
terms of consistency strength and implicational strength. I also analyze the
superstrong cardinals, which are weaker than supercompact cardinals but are
related to high-jump cardinals. Two of my most important results are as
follows. \begin{itemize} \item \Vopenka\ cardinals are the same as
Woodin-for-supercompactness cardinals. \item There are no excessively
hypercompact cardinals. \end{itemize} Furthermore, I prove some results
relating high-jump cardinals to forcing, as well as analyzing Laver functions
for super-high-jump cardinals. \keywords{high-jump cardinals \and \Vopenka\
cardinals \and Woodin-for-supercompactness cardinals \and hypercompact
cardinals \and forcing and large cardinals \and Laver functions}
