  'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in
symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on
moduli spaces of $J$-holomorphic curves. An alternative to Kuranishi spaces is
the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185).
Finding a satisfactory definition of Kuranishi space has been the subject of
recent debate (see e.g. arXiv:1208.1340, arXiv:1209.4410, arXiv:1510.06849).
  We propose three new definitions of Kuranishi space: a simple 'manifold'
version, '$\mu$-Kuranishi spaces', which form an ordinary category
$\boldsymbol\mu\bf Kur$; a more complicated 'manifold' version, 'm-Kuranishi
spaces', which form a weak 2-category $\bf mKur$; and an 'orbifold' version,
'Kuranishi spaces', which form a weak 2-category $\bf Kur$. These are related
by an equivalence of categories $\boldsymbol\mu{\bf Kur}\simeq{\rm Ho}({\bf
mKur})$, where ${\rm Ho}({\bf mKur})$ is the homotopy category of $\bf mKur$,
and by a full and faithful embedding ${\bf mKur}\hookrightarrow\bf Kur$. We
also define ($\mu$-, m-)Kuranishi spaces with boundary, and with corners. We
hope our definitions will become accepted as final, replacing previous
definitions.
  Any Fukaya-Oh-Ohta-Ono Kuranishi space $\bf X$ can be made into a compact
Kuranishi space $\bf X'$ uniquely up to equivalence in $\bf Kur$ (that is, up
to isomorphism in ${\rm Ho}({\bf Kur})$). The same holds for topological spaces
with Fukaya-Oh-Ohta-Ono 'good coordinate systems', and for McDuff and
Wehrheim's 'Kuranishi atlases' in arXiv:1508.01556. A compact topological space
$\bf X$ with a 'polyfold Fredholm structure' in the sense of Hofer, Wysocki and
Zehnder can be made into a Kuranishi space $\bf X$ uniquely up to equivalence
in $\bf Kur$.
  This book is surveyed in arXiv:1510.07444.
