In a previous article [math.CO/9712207], we derived the alternating-sign matrix
(ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice
with domain wall boundary. Here we show that the same argument enumerates three other
symmetry classes of alternating-sign matrices: VSASMs (vertically symmetric ASMs), even
HTSASMs (half-turn-symmetric ASMs), and even QTSASMs (quarter-turn-symmetric ASMs). The
VSASM enumeration was conjectured by Mills; the others by Robbins [math.CO/0008045]. We
introduce several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn
sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally, off-antidiagonally
symmetric), and UOSASMs (off-diagonally symmetric with U-turn sides). UASMs generalize
VSASMs, while UUASMs generalize VHSASMs (vertically and horizontally symmetric ASMs) and
another new class, VHPASMs (vertically and horizontally perverse). OSASMs, OOSASMs, and
UOSASMs are related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally
symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs (totally symmetric
ASMs). We enumerate several of these new classes, and we provide several 2-enumerations and
3-enumerations. Our main technical tool is a set of multi-parameter determinant and
Pfaffian formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya
determinant for UASMs [solv-int/9804010]. We evaluate specializations of the determinants
and Pfaffians using the factor exhaustion method.