  This is an introduction to finite simple groups, in particular sporadic
groups, intended for physicists. After a short review of group theory, we
enumerate the $1+1+16=18$ families of finite simple groups, as an introduction
to the sporadic groups. These are described next, in three levels of increasing
complexity, plus the six isolated "pariah" groups. The (old) five Mathieu
groups make up the first, smallest order level. The seven groups related to the
Leech lattice, including the three Conway groups, constitute the second level.
The third and highest level contains the Monster group $\mathbb M$, plus seven
other related groups. Next a brief mention is made of the remaining six pariah
groups, thus completing the $5+7+8+6=26$ sporadic groups. The review ends up
with a brief discussion of a few of physical applications of finite groups in
physics, including a couple of recent examples which use sporadic groups.
