diff --git a/lib/matplotlib/text.py b/lib/matplotlib/text.py
index acde4fb179a2..31be2cbdf04e 100644
--- a/lib/matplotlib/text.py
+++ b/lib/matplotlib/text.py
@@ -1625,12 +1625,41 @@ def _check_xy(self, renderer=None):
         if renderer is None:
             renderer = self.get_figure(root=True)._get_renderer()
         b = self.get_annotation_clip()
+        not_none = (b is None and
+                    callable(self.xycoords) and
+                    self.xycoords(renderer) is not None)
         if b or (b is None and self.xycoords == "data"):
             # check if self.xy is inside the Axes.
             xy_pixel = self._get_position_xy(renderer)
             return self.axes.contains_point(xy_pixel)
+
+        if not_none:
+            if self.xycoords(renderer).is_null():
+                return False
+
         return True
 
+    def _check_xytext(self, renderer=None):
+        """Check whether the annotation text at *xy_pixel* can be drawn."""
+        valid = True
+
+        if self._xytext is None:
+            return valid
+
+        # transforming the coordinates
+        x = self.convert_xunits(self._xytext[0])
+        y = self.convert_yunits(self._xytext[1])
+        unitless_coords = (x, y)
+        coords = np.array(self.get_transform().transform(unitless_coords))
+        valid = not np.isnan(coords).any() and np.isfinite(coords).all()
+        # DEBUG
+        print("###")
+        print(coords)
+
+        if not valid:
+            raise ValueError("xytext coordinates must be finite numbers")
+        return valid
+
     def draggable(self, state=None, use_blit=False):
         """
         Set whether the annotation is draggable with the mouse.
@@ -1862,6 +1891,7 @@ def transform(renderer) -> Transform
                                  xy,
                                  xycoords=xycoords,
                                  annotation_clip=annotation_clip)
+        self._xytext = xytext
         # warn about wonky input data
         if (xytext is None and
                 textcoords is not None and
@@ -2027,11 +2057,16 @@ def draw(self, renderer):
         # docstring inherited
         if renderer is not None:
             self._renderer = renderer
-        if not self.get_visible() or not self._check_xy(renderer):
+
+        visible = self.get_visible() and self._check_xy(renderer)
+
+
+        if not visible:
             return
         # Update text positions before `Text.draw` would, so that the
         # FancyArrowPatch is correctly positioned.
         self.update_positions(renderer)
+        self._check_xytext()
         self.update_bbox_position_size(renderer)
         if self.arrow_patch is not None:  # FancyArrowPatch
             if (self.arrow_patch.get_figure(root=False) is None and
@@ -2046,7 +2081,10 @@ def get_window_extent(self, renderer=None):
         # docstring inherited
         # This block is the same as in Text.get_window_extent, but we need to
         # set the renderer before calling update_positions().
-        if not self.get_visible() or not self._check_xy(renderer):
+        visible = self.get_visible() and self._check_xy(renderer)
+
+
+        if not visible:
             return Bbox.unit()
         if renderer is not None:
             self._renderer = renderer
@@ -2056,6 +2094,7 @@ def get_window_extent(self, renderer=None):
             raise RuntimeError('Cannot get window extent without renderer')
 
         self.update_positions(self._renderer)
+        self._check_xytext()
 
         text_bbox = Text.get_window_extent(self)
         bboxes = [text_bbox]
diff --git a/lib/matplotlib/transforms.py b/lib/matplotlib/transforms.py
index 350113c56170..774bfceabadf 100644
--- a/lib/matplotlib/transforms.py
+++ b/lib/matplotlib/transforms.py
@@ -783,6 +783,14 @@ def frozen(self):
         frozen_bbox._minpos = self.minpos.copy()
         return frozen_bbox
 
+    def is_null(self):
+        """
+        Return True if this bbox is a 'null' bbox, i.e.
+        [[inf, inf], [-inf, -inf]].
+        """
+        return (np.isposinf(self.x0) and np.isposinf(self.y0) and
+                np.isneginf(self.x1) and np.isneginf(self.y1))
+
     @staticmethod
     def unit():
         """Create a new unit `Bbox` from (0, 0) to (1, 1)."""