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A smooth complete irreducible [[Algebraic variety|algebraic variety]]  $  X $
over a field  $  k $
whose anti-canonical sheaf  $  K _ {X}  ^ {-1} $
is ample (cf. [[Ample sheaf|Ample sheaf]]). The basic research into such varieties was done by G. Fano ([[#References|[1]]], [[#References|[2]]]).

A Fano variety of dimension 2 is called a del Pezzo surface and is a [[Rational surface|rational surface]]. The multi-dimensional analogues of del Pezzo surfaces — Fano varieties of dimension  $  > 2 $— 
are not all rational varieties, for example the general cubic in the projective space  $  P  ^ {4} $. 
It is not known (1984) whether all Fano varieties are unirational.

Three-dimensional Fano varieties have been thoroughly investigated (see [[#References|[3]]], ). Only isolated particular results are known about Fano varieties of dimension greater than 3.

The [[Picard group|Picard group]]  $   \mathop{\rm Pic}  X $
of a three-dimensional Fano variety is finitely generated and torsion-free. In case the ground field  $  k $
is  $  \mathbf C $, 
the rank of  $   \mathop{\rm Pic}  X $, 
which is equal to the second [[Betti number|Betti number]]  $  b _ {2} ( X) $, 
does not exceed 10 (see [[#References|[4]]]). If  $  b _ {2} ( X) \geq  6 $, 
then the Fano variety is isomorphic to  $  P  ^ {1} \times S _ {11 - b _ {2}  ( X) } $, 
where  $  S _ {d} $
is the del Pezzo surface of order  $  d $. 
A Fano variety  $  X $
is called primitive if there is no [[Monoidal transformation|monoidal transformation]]  $  \sigma :  X \rightarrow X _ {1} $
to a smooth variety  $  X _ {1} $
with centre at a non-singular irreducible curve. If  $  X $
is a primitive Fano variety, then  $  b _ {2} ( X) \leq  3 $. 
If  $  b _ {2} ( X) = 3 $, 
then  $  X $
is a conic fibre space over  $  S = P  ^ {1} \times P  ^ {1} $, 
in other words, then there is a morphism  $  \pi :  X \rightarrow S $
each fibre of which is isomorphic to a conic, that is, an algebraic scheme given by a homogeneous equation of degree 2 in  $  P  ^ {2} $. 
A Fano variety  $  X $
with  $  b _ {2} ( X) = 2 $
is a conic fibre space over the projective plane  $  P  ^ {2} $(
see [[#References|[3]]]). In the case  $  b _ {2} ( X) = 1 $
there are 18 types of Fano varieties, which have been classified (see [[#References|[6]]]).

For three-dimensional Fano varieties  $  X $
the self-intersection index of the anti-canonical divisor  $  (- K _ {X}  ^ {3} ) \leq  64 $. 
The largest integer  $  r \geq  1 $
such that  $  H ^ {\otimes r } $
is isomorphic to  $  K _ {X}  ^ {-1} $
for some divisor  $  H \in  \mathop{\rm Pic}  X $
is called the index of the Fano variety. The index of a three-dimensional Fano variety can take the values 1, 2, 3, or 4. A Fano variety of index 4 is isomorphic to the projective space  $  P  ^ {3} $, 
and a Fano variety of index 3 is isomorphic to a smooth quadric  $  Q \subset  P  ^ {4} $. 
If  $  r = 2 $, 
then the self-intersection index  $  d = H  ^ {3} $
can take the values  $  1 \leq  d \leq  7 $, 
with each of them being realized for some Fano variety. For a Fano variety of index 1 the mapping  $  \phi _ {K _ {X}   ^ {-1} } :  X \rightarrow P ^ { \mathop{\rm dim}  | K _ {X}  ^ {-1} | } $
defined by the linear system  $  | K _ {X}  ^ {-1} | $
has degree  $   \mathop{\rm deg}  \phi _ {K _ {X}   ^ {-1} } = 1 $
or 2. The Fano varieties of index 1 for which  $   \mathop{\rm deg}  \phi _ {K _ {X}   ^ {-1} } = 2 $
have been classified. If  $   \mathop{\rm deg}  \phi _ {K _ {X}   ^ {-1} } = 1 $, 
then  $  X $
can be realized as a subvariety  $  V _ {2g - 2 }  $
of degree  $  2g - 2 $
in the projective space  $  P ^ {g + 1 } $. 
The number  $  g $
is called the genus of the Fano variety  $  V _ {2g - 2 }  $
and is the same as the genus of the canonical curve — the section of  $  X $
under the anti-canonical imbedding into  $  P ^ {g + 1 } $. 
The Fano varieties  $  V _ {2g - 2 }  \subset  P ^ {g + 1 } $
the class of a hyperplane section of which is the same as the anti-canonical class and generates  $   \mathop{\rm Pic}  V _ {2g - 2 }  $
have been classified (see [[#References|[4]]], ).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  G. Fano,   "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu" , ''Proc. Internat. Congress Mathematicians (Bologna)'' , '''4''' , Zanichelli  (1934)  pp. 115–119</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  G. Fano,   "Si alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche"  ''Comment. Math. Helv.'' , '''14'''  (1942)  pp. 202–211</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Mori,   S. Mukai,   "Classification of Fano 3-folds with $B_2 \geq 2$"  ''Manuscripta Math.'' , '''36''' :  2  (1981)  pp. 147–162</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  L. Roth,   "Sulle $V_3$ algebriche su cui l'aggiunzione si estingue"  ''Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat.'' , '''9'''  (1950)  pp. 246–250</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  V.A. Iskovskikh,   "Fano 3-folds. I"  ''Math. USSR. Izv.'' , '''11''' :  3  (1977)  pp. 485–527  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''41''' :  3  (1977)  pp. 516–562</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  V.A. Iskovskikh,   "Fano 3-folds. II"  ''Math. USSR. Izv.'' , '''12''' :  3  (1978)  pp. 469–506  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''42''' :  3  (1978)  pp. 506–549</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.A. Iskovskikh,   "Anticanonical models of three-dimensional algebraic varieties"  ''J. Soviet Math.'' , '''13''' :  6  (1980)  pp. 745–850  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''12'''  (1979)  pp. 59–157</TD></TR>
</table>