diff --git a/doc/api/axes_api.rst b/doc/api/axes_api.rst
index f78f4c062c06..84817c6d6bba 100644
--- a/doc/api/axes_api.rst
+++ b/doc/api/axes_api.rst
@@ -87,6 +87,7 @@ Spans
    Axes.axhspan
    Axes.axvline
    Axes.axvspan
+   Axes.axline
 
 Spectral
 --------
diff --git a/doc/users/next_whats_new/2017-12-08-axline.rst b/doc/users/next_whats_new/2017-12-08-axline.rst
new file mode 100644
index 000000000000..f4ba4e0b3fa9
--- /dev/null
+++ b/doc/users/next_whats_new/2017-12-08-axline.rst
@@ -0,0 +1,5 @@
+New `~.axes.Axes.axline` method
+-------------------------------
+
+A new `~.axes.Axes.axline` method has been added to draw infinitely long lines
+that pass through two points.
diff --git a/examples/subplots_axes_and_figures/axhspan_demo.py b/examples/subplots_axes_and_figures/axhspan_demo.py
index ab1164076e7e..0cc9b49dd22b 100644
--- a/examples/subplots_axes_and_figures/axhspan_demo.py
+++ b/examples/subplots_axes_and_figures/axhspan_demo.py
@@ -4,32 +4,33 @@
 ============
 
 Create lines or rectangles that span the axes in either the horizontal or
-vertical direction.
+vertical direction, and lines than span the axes with an arbitrary orientation.
 """
+
 import numpy as np
 import matplotlib.pyplot as plt
 
 t = np.arange(-1, 2, .01)
 s = np.sin(2 * np.pi * t)
 
-plt.plot(t, s)
-# Draw a thick red hline at y=0 that spans the xrange
-plt.axhline(linewidth=8, color='#d62728')
-
-# Draw a default hline at y=1 that spans the xrange
-plt.axhline(y=1)
-
-# Draw a default vline at x=1 that spans the yrange
-plt.axvline(x=1)
-
-# Draw a thick blue vline at x=0 that spans the upper quadrant of the yrange
-plt.axvline(x=0, ymin=0.75, linewidth=8, color='#1f77b4')
-
-# Draw a default hline at y=.5 that spans the middle half of the axes
-plt.axhline(y=.5, xmin=0.25, xmax=0.75)
-
-plt.axhspan(0.25, 0.75, facecolor='0.5', alpha=0.5)
-
-plt.axvspan(1.25, 1.55, facecolor='#2ca02c', alpha=0.5)
+fig, ax = plt.subplots()
+
+ax.plot(t, s)
+# Thick red horizontal line at y=0 that spans the xrange.
+ax.axhline(linewidth=8, color='#d62728')
+# Horizontal line at y=1 that spans the xrange.
+ax.axhline(y=1)
+# Vertical line at x=1 that spans the yrange.
+ax.axvline(x=1)
+# Thick blue vertical line at x=0 that spans the upper quadrant of the yrange.
+ax.axvline(x=0, ymin=0.75, linewidth=8, color='#1f77b4')
+# Default hline at y=.5 that spans the middle half of the axes.
+ax.axhline(y=.5, xmin=0.25, xmax=0.75)
+# Infinite black line going through (0, 0) to (1, 1).
+ax.axline((0, 0), (1, 1), color='k')
+# 50%-gray rectangle spanning the axes' width from y=0.25 to y=0.75.
+ax.axhspan(0.25, 0.75, facecolor='0.5')
+# Green rectangle spanning the axes' height from x=1.25 to x=1.55.
+ax.axvspan(1.25, 1.55, facecolor='#2ca02c')
 
 plt.show()
diff --git a/lib/matplotlib/axes/_axes.py b/lib/matplotlib/axes/_axes.py
index 9d609aa9891b..83b67733b993 100644
--- a/lib/matplotlib/axes/_axes.py
+++ b/lib/matplotlib/axes/_axes.py
@@ -816,6 +816,7 @@ def axhline(self, y=0, xmin=0, xmax=1, **kwargs):
         --------
         hlines : Add horizontal lines in data coordinates.
         axhspan : Add a horizontal span (rectangle) across the axis.
+        axline : Add a line with an arbitrary slope.
 
         Examples
         --------
@@ -899,6 +900,7 @@ def axvline(self, x=0, ymin=0, ymax=1, **kwargs):
         --------
         vlines : Add vertical lines in data coordinates.
         axvspan : Add a vertical span (rectangle) across the axis.
+        axline : Add a line with an abritrary slope.
         """
 
         if "transform" in kwargs:
@@ -919,6 +921,63 @@ def axvline(self, x=0, ymin=0, ymax=1, **kwargs):
         self._request_autoscale_view(scalex=scalex, scaley=False)
         return l
 
+    @docstring.dedent_interpd
+    def axline(self, xy1, xy2, **kwargs):
+        """
+        Add an infinitely long straight line that passes through two points.
+
+        This draws a straight line "on the screen", regardless of the x and y
+        scales, and is thus also suitable for drawing exponential decays in
+        semilog plots, power laws in loglog plots, etc.
+
+        Parameters
+        ----------
+        xy1, xy2 : (float, float)
+            Points for the line to pass through.
+
+        Returns
+        -------
+        :class:`~matplotlib.lines.Line2D`
+
+        Other Parameters
+        ----------------
+        **kwargs
+            Valid kwargs are :class:`~matplotlib.lines.Line2D` properties,
+            with the exception of 'transform':
+
+            %(_Line2D_docstr)s
+
+        Examples
+        --------
+        Draw a thick red line passing through (0, 0) and (1, 1)::
+
+            >>> axline((0, 0), (1, 1), linewidth=4, color='r')
+
+        See Also
+        --------
+        axhline : for horizontal lines
+        axvline : for vertical lines
+        """
+
+        if "transform" in kwargs:
+            raise TypeError("'transform' is not allowed as a kwarg; "
+                            "axline generates its own transform")
+        x1, y1 = xy1
+        x2, y2 = xy2
+        line = mlines._AxLine([x1, x2], [y1, y2], **kwargs)
+        # Like add_line, but correctly handling data limits.
+        self._set_artist_props(line)
+        if line.get_clip_path() is None:
+            line.set_clip_path(self.patch)
+        if not line.get_label():
+            line.set_label(f"_line{len(self.lines)}")
+        self.lines.append(line)
+        line._remove_method = self.lines.remove
+        self.update_datalim([xy1, xy2])
+
+        self._request_autoscale_view()
+        return line
+
     @docstring.dedent_interpd
     def axhspan(self, ymin, ymax, xmin=0, xmax=1, **kwargs):
         """
diff --git a/lib/matplotlib/lines.py b/lib/matplotlib/lines.py
index c67c6f35f8e1..6cad1b744d13 100644
--- a/lib/matplotlib/lines.py
+++ b/lib/matplotlib/lines.py
@@ -15,7 +15,8 @@
     _to_unmasked_float_array, ls_mapper, ls_mapper_r, STEP_LOOKUP_MAP)
 from .markers import MarkerStyle
 from .path import Path
-from .transforms import Bbox, TransformedPath
+from .transforms import (
+    Affine2D, Bbox, BboxTransformFrom, BboxTransformTo, TransformedPath)
 
 # Imported here for backward compatibility, even though they don't
 # really belong.
@@ -1448,6 +1449,47 @@ def is_dashed(self):
         return self._linestyle in ('--', '-.', ':')
 
 
+class _AxLine(Line2D):
+    """
+    A helper class that implements `~.Axes.axline`, by recomputing the artist
+    transform at draw time.
+    """
+
+    def get_transform(self):
+        ax = self.axes
+        (x1, y1), (x2, y2) = ax.transScale.transform([*zip(*self.get_data())])
+        dx = x2 - x1
+        dy = y2 - y1
+        if np.allclose(x1, x2):
+            if np.allclose(y1, y2):
+                raise ValueError(
+                    f"Cannot draw a line through two identical points "
+                    f"(x={self.get_xdata()}, y={self.get_ydata()})")
+            # First send y1 to 0 and y2 to 1.
+            return (Affine2D.from_values(1, 0, 0, 1 / dy, 0, -y1 / dy)
+                    + ax.get_xaxis_transform(which="grid"))
+        if np.allclose(y1, y2):
+            # First send x1 to 0 and x2 to 1.
+            return (Affine2D.from_values(1 / dx, 0, 0, 1, -x1 / dx, 0)
+                    + ax.get_yaxis_transform(which="grid"))
+        (vxlo, vylo), (vxhi, vyhi) = ax.transScale.transform(ax.viewLim)
+        # General case: find intersections with view limits in either
+        # direction, and draw between the middle two points.
+        _, start, stop, _ = sorted([
+            (vxlo, y1 + (vxlo - x1) * dy / dx),
+            (vxhi, y1 + (vxhi - x1) * dy / dx),
+            (x1 + (vylo - y1) * dx / dy, vylo),
+            (x1 + (vyhi - y1) * dx / dy, vyhi),
+        ])
+        return (BboxTransformFrom(Bbox([*zip(*self.get_data())]))
+                + BboxTransformTo(Bbox([start, stop]))
+                + ax.transLimits + ax.transAxes)
+
+    def draw(self, renderer):
+        self._transformed_path = None  # Force regen.
+        super().draw(renderer)
+
+
 class VertexSelector:
     """
     Manage the callbacks to maintain a list of selected vertices for
diff --git a/lib/matplotlib/pyplot.py b/lib/matplotlib/pyplot.py
index 6da0ed1fab93..9e03039d959f 100644
--- a/lib/matplotlib/pyplot.py
+++ b/lib/matplotlib/pyplot.py
@@ -2371,6 +2371,12 @@ def axis(*args, emit=True, **kwargs):
     return gca().axis(*args, emit=emit, **kwargs)
 
 
+# Autogenerated by boilerplate.py.  Do not edit as changes will be lost.
+@docstring.copy(Axes.axline)
+def axline(xy1, xy2, **kwargs):
+    return gca().axline(xy1, xy2, **kwargs)
+
+
 # Autogenerated by boilerplate.py.  Do not edit as changes will be lost.
 @docstring.copy(Axes.axvline)
 def axvline(x=0, ymin=0, ymax=1, **kwargs):
diff --git a/lib/matplotlib/tests/test_axes.py b/lib/matplotlib/tests/test_axes.py
index d9e6924dd3bd..16f495c04375 100644
--- a/lib/matplotlib/tests/test_axes.py
+++ b/lib/matplotlib/tests/test_axes.py
@@ -3641,6 +3641,21 @@ def test_eb_line_zorder():
     ax.set_title("errorbar zorder test")
 
 
+@check_figures_equal()
+def test_axline(fig_test, fig_ref):
+    ax = fig_test.subplots()
+    ax.set(xlim=(-1, 1), ylim=(-1, 1))
+    ax.axline((0, 0), (1, 1))
+    ax.axline((0, 0), (1, 0), color='C1')
+    ax.axline((0, 0.5), (1, 0.5), color='C2')
+
+    ax = fig_ref.subplots()
+    ax.set(xlim=(-1, 1), ylim=(-1, 1))
+    ax.plot([-1, 1], [-1, 1])
+    ax.axhline(0, color='C1')
+    ax.axhline(0.5, color='C2')
+
+
 @image_comparison(['vlines_basic', 'vlines_with_nan', 'vlines_masked'],
                   extensions=['png'])
 def test_vlines():
diff --git a/tools/boilerplate.py b/tools/boilerplate.py
index 5bb4068a42ed..cafb850b44af 100644
--- a/tools/boilerplate.py
+++ b/tools/boilerplate.py
@@ -194,6 +194,7 @@ def boilerplate_gen():
         'axhline',
         'axhspan',
         'axis',
+        'axline',
         'axvline',
         'axvspan',
         'bar',