diff --git a/babel/dates.py b/babel/dates.py
index 7b626aa11..b83506b84 100644
--- a/babel/dates.py
+++ b/babel/dates.py
@@ -1480,7 +1480,7 @@ def __getitem__(self, name: str) -> str:
         else:
             raise KeyError(f"Unsupported date/time field {char!r}")
 
-    def extract(self, char: str) -> int:
+    def extract(self, char: str) -> int | str:
         char = str(char)[0]
         if char == 'y':
             return self.value.year
@@ -1496,6 +1496,12 @@ def extract(self, char: str) -> int:
             return self.value.minute
         elif char == 'a':
             return int(self.value.hour >= 12)  # 0 for am, 1 for pm
+        elif char == 'G':
+            return int(self.value.year >= 0)  # 0 for BCE, 1 for CE
+        elif char == 'B':
+            # The day period is rendered from the same identifier,
+            # so the extracted value matches what `format_period` renders.
+            return get_period_id(self.value, locale=self.locale)
         else:
             raise NotImplementedError(
                 f"Not implemented: extracting {char!r} from {self.value!r}",
@@ -1621,14 +1627,22 @@ def format_period(self, char: str, num: int) -> str:
         ]
         if char == 'a':
             period = 'pm' if self.value.hour >= 12 else 'am'
-            context = 'format'
+            contexts = ('format',)
         else:
             period = get_period_id(self.value, locale=self.locale)
-            context = 'format' if char == 'B' else 'stand-alone'
-        for width in widths:
-            period_names = get_period_names(context=context, width=width, locale=self.locale)
-            if period in period_names:
-                return period_names[period]
+            # CLDR falls back from a stand-alone name to the format name
+            contexts = ('format', 'stand-alone') if char == 'B' else ('stand-alone', 'format')
+        # A locale without a name for the flexible day period falls back to
+        # the name of the day period it belongs to, as CLDR does
+        general_period = 'pm' if self.value.hour >= 12 else 'am'
+        for candidate in (period, general_period):
+            for context in contexts:
+                for width in widths:
+                    period_names = get_period_names(
+                        context=context, width=width, locale=self.locale,
+                    )
+                    if candidate in period_names:
+                        return period_names[candidate]
         raise ValueError(f"Could not format period {period} in {self.locale}")
 
     def format_frac_seconds(self, num: int) -> str:
diff --git a/tests/test_date_intervals.py b/tests/test_date_intervals.py
index 33a85d38a..ee52b75c1 100644
--- a/tests/test_date_intervals.py
+++ b/tests/test_date_intervals.py
@@ -61,3 +61,32 @@ def test_issue_825():
         locale='pt',
     )
     assert formatted == '1 h'
+
+
+@pytest.mark.parametrize(
+    ("start", "end", "skeleton", "expected"),
+    [
+        # Era ('G') interval formats used to raise NotImplementedError when
+        # computing the greatest difference between the two instants.
+        # A year difference must select the 'y' pattern of the "Gy" interval format.
+        (TEST_DT, TEST_DT.replace(year=2017), "Gy", "2016\u2009\u2013\u20092017 AD"),
+        (
+            TEST_DT,
+            TEST_DT.replace(year=2017, month=3),
+            "GyM",
+            "1/2016\u2009\u2013\u20093/2017 AD",
+        ),
+        # Day period ('B') interval formats used to raise NotImplementedError, too.
+        # Differing day periods must select the 'B' pattern of the "Bh" interval format.
+        (
+            TEST_DT,
+            TEST_DT.replace(hour=16),
+            "Bh",
+            "11 in the morning\u2009\u2013\u20094 in the afternoon",
+        ),
+        # ... and a shared day period must fall through to the 'h' pattern.
+        (TEST_DT, TEST_DT.replace(hour=10), "Bh", "11\u2009\u2013\u200910 in the morning"),
+    ],
+)
+def test_format_interval_era_and_day_period(start, end, skeleton, expected):
+    assert dates.format_interval(start, end, skeleton, locale="en") == expected
diff --git a/tests/test_dates.py b/tests/test_dates.py
index 4f8c73907..222bebda6 100644
--- a/tests/test_dates.py
+++ b/tests/test_dates.py
@@ -203,6 +203,18 @@ def test_get_period_names():
     assert dates.get_period_names(locale='en_US')['am'] == 'AM'
 
 
+def test_format_period_without_names_for_the_flexible_day_period():
+    # Locales such as az_Arab only carry names for 'am'/'pm', so the flexible
+    # day period that 'B' and 'b' name has no name to return.  Fall back to
+    # the name of the day period the time belongs to instead of raising.
+    dt = datetime(2020, 1, 1, 11, 30)
+    assert dates.format_datetime(dt, 'B', locale='az_Arab') == 'AM'
+    assert dates.format_datetime(dt, 'b', locale='az_Arab') == 'AM'
+    assert dates.format_datetime(dt, 'a', locale='az_Arab') == 'AM'
+    # A locale that does name the flexible day period is unaffected.
+    assert dates.format_datetime(dt, 'B', locale='fi_FI') == 'aamup.'
+
+
 def test_get_day_names():
     assert dates.get_day_names('wide', locale='en_US')[1] == 'Tuesday'
     assert dates.get_day_names('short', locale='en_US')[1] == 'Tu'