diff --git a/src/doc/en/reference/references/index.rst b/src/doc/en/reference/references/index.rst
index a2010e67e33..9c4192205cf 100644
--- a/src/doc/en/reference/references/index.rst
+++ b/src/doc/en/reference/references/index.rst
@@ -1658,9 +1658,12 @@ REFERENCES:
              constrained optimization. ACM Transactions on
              Mathematical Software, Vol 23, Num. 4, pp.550--560, 1997.
 
-.. [Zie1998] G. Ziegler. *Shelling polyhedral 3-balls and
+.. [Zie1998] G. M. Ziegler. *Shelling polyhedral 3-balls and
              4-polytopes*. Discrete Comput. Geom. 19 (1998), 159-174.
 
+.. [Zie2007] G. M. Ziegler. *Lectures on polytopes*, Volume
+             152 of Graduate Texts in Mathematics, 7th printing of 1st edition, Springer, 2007.
+
 .. [Zor2008] \A. Zorich "Explicit Jenkins-Strebel representatives of
              all strata of Abelian and quadratic differentials",
              Journal of Modern Dynamics, vol. 2, no 1, 139-185 (2008)
diff --git a/src/sage/geometry/polyhedron/base.py b/src/sage/geometry/polyhedron/base.py
index 50d99f99e46..cb85668b3a4 100644
--- a/src/sage/geometry/polyhedron/base.py
+++ b/src/sage/geometry/polyhedron/base.py
@@ -464,7 +464,7 @@ def _is_subpolyhedron(self, other):
                     for self_V in self.Vrepresentation())
 
     def plot(self,
-             point=None, line=None, polygon=None, #  None means unspecified by the user
+             point=None, line=None, polygon=None,  # None means unspecified by the user
              wireframe='blue', fill='green',
              projection_direction=None,
              **kwds):
@@ -1554,9 +1554,9 @@ def vertices_matrix(self, base_ring=None):
         if base_ring is None:
             base_ring = self.base_ring()
         m = matrix(base_ring, self.ambient_dim(), self.n_vertices())
-        for i,v in enumerate(self.vertices()):
+        for i, v in enumerate(self.vertices()):
             for j in range(self.ambient_dim()):
-                m[j,i] = v[j]
+                m[j, i] = v[j]
         return m
 
     def ray_generator(self):
@@ -2390,6 +2390,114 @@ def gale_transform(self):
         A_ker = A.right_kernel()
         return A_ker.basis_matrix().transpose().rows()
 
+    @cached_method
+    def normal_fan(self):
+        r"""
+        Return the normal fan of a compact full-dimensional rational polyhedron.
+
+        OUTPUT:
+
+        A complete fan of the ambient space as a
+        :class:`~sage.geometry.fan.RationalPolyhedralFan`.
+
+        .. SEEALSO::
+
+            :meth:`~sage.geometry.polyhedron.base.face_fan`.
+
+        EXAMPLES::
+
+            sage: S = Polyhedron(vertices = [[0, 0], [1, 0], [0, 1]])
+            sage: S.normal_fan()
+            Rational polyhedral fan in 2-d lattice N
+
+            sage: C = polytopes.hypercube(4)
+            sage: NF = C.normal_fan(); NF
+            Rational polyhedral fan in 4-d lattice N
+
+        Currently, it is only possible to get the normal fan of a bounded rational polytope::
+
+            sage: P = Polyhedron(rays = [[1, 0], [0, 1]])
+            sage: P.normal_fan()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: the normal fan is only supported for polytopes (compact polyhedra).
+
+            sage: Q = Polyhedron(vertices = [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+            sage: Q.normal_fan()
+            Traceback (most recent call last):
+            ...
+            ValueError: the normal fan is only defined for full-dimensional polytopes
+
+            sage: R = Polyhedron(vertices = [[0, 0], [AA(sqrt(2)), 0], [0, AA(sqrt(2))]])
+            sage: R.normal_fan()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: normal fan handles only polytopes over the rationals
+
+        REFERENCES:
+
+        For more information, see Chapter 7 of [Zie2007]_.
+        """
+        from sage.geometry.fan import NormalFan
+
+        if not QQ.has_coerce_map_from(self.base_ring()):
+            raise NotImplementedError('normal fan handles only polytopes over the rationals')
+
+        return NormalFan(self)
+
+    @cached_method
+    def face_fan(self):
+        r"""
+        Return the face fan of a compact rational polyhedron.
+
+        OUTPUT:
+
+        A fan of the ambient space as a
+        :class:`~sage.geometry.fan.RationalPolyhedralFan`.
+
+        .. SEEALSO::
+
+            :meth:`~sage.geometry.polyhedron.base.normal_fan`.
+
+        EXAMPLES::
+
+            sage: T = polytopes.cuboctahedron()
+            sage: T.face_fan()
+            Rational polyhedral fan in 3-d lattice M
+
+        The polytope should contain the origin in the interior::
+
+            sage: P = Polyhedron(vertices = [[1/2, 1], [1, 1/2]])
+            sage: P.face_fan()
+            Traceback (most recent call last):
+            ...
+            ValueError: face fans are defined only for polytopes containing the origin as an interior point!
+
+            sage: Q = Polyhedron(vertices = [[-1, 1/2], [1, -1/2]])
+            sage: Q.contains([0,0])
+            True
+            sage: FF = Q.face_fan(); FF
+            Rational polyhedral fan in 2-d lattice M
+
+        The polytope has to have rational coordinates::
+
+            sage: S = polytopes.dodecahedron()
+            sage: S.face_fan()
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: face fan handles only polytopes over the rationals
+
+        REFERENCES:
+
+        For more information, see Chapter 7 of [Zie2007]_.
+        """
+        from sage.geometry.fan import FaceFan
+
+        if not QQ.has_coerce_map_from(self.base_ring()):
+            raise NotImplementedError('face fan handles only polytopes over the rationals')
+
+        return FaceFan(self)
+
     def triangulate(self, engine='auto', connected=True, fine=False, regular=None, star=None):
         r"""
         Returns a triangulation of the polytope.