diff --git a/containers-tests/tests/intset-properties.hs b/containers-tests/tests/intset-properties.hs
index c93d6e8af..201a27104 100644
--- a/containers-tests/tests/intset-properties.hs
+++ b/containers-tests/tests/intset-properties.hs
@@ -7,6 +7,11 @@ import Data.List (nub,sort)
 import qualified Data.List as List
 import Data.Maybe (listToMaybe)
 import Data.Monoid (mempty)
+#if MIN_VERSION_base(4,18,0)
+import Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NE
+import qualified Data.Foldable1 as Foldable1
+#endif
 import qualified Data.Set as Set
 import IntSetValidity (valid)
 import Prelude hiding (lookup, null, map, filter, foldr, foldl, foldl')
@@ -82,6 +87,10 @@ main = defaultMain $ testGroup "intset-properties"
                    , testProperty "prop_bitcount" prop_bitcount
                    , testProperty "prop_alterF_list" prop_alterF_list
                    , testProperty "prop_alterF_const" prop_alterF_const
+#if MIN_VERSION_base(4,18,0)
+                   , testProperty "intersections" prop_intersections
+                   , testProperty "intersections_lazy" prop_intersections_lazy
+#endif
                    ]
 
 ----------------------------------------------------------------
@@ -500,3 +509,18 @@ prop_alterF_const
 prop_alterF_const f k s =
         getConst (alterF     (Const . applyFun f) k s        )
     === getConst (Set.alterF (Const . applyFun f) k (toSet s))
+
+#if MIN_VERSION_base(4,18,0)
+prop_intersections :: (IntSet, [IntSet]) -> Property
+prop_intersections (s, ss) =
+  intersections ss' === Foldable1.foldl1' intersection ss'
+  where
+    ss' = s :| ss -- Work around missing Arbitrary NonEmpty instance
+
+prop_intersections_lazy :: [IntSet] -> Property
+prop_intersections_lazy ss = intersections ss' === empty
+  where
+    ss' = NE.fromList $ ss ++ [empty] ++ undefined
+                            -- ^ result will certainly be empty at this point,
+                            --   so the rest of the list should not be demanded.
+#endif
diff --git a/containers-tests/tests/set-properties.hs b/containers-tests/tests/set-properties.hs
index 82d30f0c6..90a3e2d5d 100644
--- a/containers-tests/tests/set-properties.hs
+++ b/containers-tests/tests/set-properties.hs
@@ -16,6 +16,11 @@ import Control.Monad (liftM, liftM3)
 import Data.Functor.Identity
 import Data.Foldable (all)
 import Control.Applicative (liftA2)
+#if MIN_VERSION_base(4,18,0)
+import Data.List.NonEmpty (NonEmpty(..))
+import qualified Data.List.NonEmpty as NE
+import qualified Data.Foldable1 as Foldable1
+#endif
 
 #if __GLASGOW_HASKELL__ >= 806
 import Utils.NoThunks (whnfHasNoThunks)
@@ -112,6 +117,10 @@ main = defaultMain $ testGroup "set-properties"
 #endif
                    , testProperty "eq" prop_eq
                    , testProperty "compare" prop_compare
+#if MIN_VERSION_base(4,18,0)
+                   , testProperty "intersections" prop_intersections
+                   , testProperty "intersections_lazy" prop_intersections_lazy
+#endif
                    ]
 
 -- A type with a peculiar Eq instance designed to make sure keys
@@ -738,3 +747,18 @@ prop_eq s1 s2 = (s1 == s2) === (toList s1 == toList s2)
 
 prop_compare :: Set Int -> Set Int -> Property
 prop_compare s1 s2 = compare s1 s2 === compare (toList s1) (toList s2)
+
+#if MIN_VERSION_base(4,18,0)
+prop_intersections :: (Set Int, [Set Int]) -> Property
+prop_intersections (s, ss) =
+  intersections ss' === Foldable1.foldl1' intersection ss'
+  where
+    ss' = s :| ss -- Work around missing Arbitrary NonEmpty instance
+
+prop_intersections_lazy :: [Set Int] -> Property
+prop_intersections_lazy ss = intersections ss' === empty
+  where
+    ss' = NE.fromList $ ss ++ [empty] ++ undefined
+                            -- ^ result will certainly be empty at this point,
+                            --   so the rest of the list should not be demanded.
+#endif
diff --git a/containers/changelog.md b/containers/changelog.md
index c0e00aa83..d650886fc 100644
--- a/containers/changelog.md
+++ b/containers/changelog.md
@@ -39,6 +39,9 @@
 
 * Add `lookupMin` and `lookupMax` for `Data.IntSet`. (Soumik Sarkar)
 
+* Add `Intersection` and `intersections` for `Data.Set` and `Data.IntSet`.
+  (Reed Mullanix, Soumik Sarkar)
+
 ## Unreleased with `@since` annotation for 0.7.1:
 
 ### Additions
diff --git a/containers/src/Data/IntSet.hs b/containers/src/Data/IntSet.hs
index 9d497e91e..83d555f2d 100644
--- a/containers/src/Data/IntSet.hs
+++ b/containers/src/Data/IntSet.hs
@@ -106,7 +106,11 @@ module Data.IntSet (
             , difference
             , (\\)
             , intersection
+#if MIN_VERSION_base(4,18,0)
+            , intersections
+#endif
             , symmetricDifference
+            , Intersection(..)
 
             -- * Filter
             , IS.filter
diff --git a/containers/src/Data/IntSet/Internal.hs b/containers/src/Data/IntSet/Internal.hs
index 0af00e4aa..f8afaed64 100644
--- a/containers/src/Data/IntSet/Internal.hs
+++ b/containers/src/Data/IntSet/Internal.hs
@@ -125,7 +125,11 @@ module Data.IntSet.Internal (
     , unions
     , difference
     , intersection
+#if MIN_VERSION_base(4,18,0)
+    , intersections
+#endif
     , symmetricDifference
+    , Intersection(..)
 
     -- * Filter
     , filter
@@ -192,11 +196,15 @@ import Control.DeepSeq (NFData(rnf))
 import Data.Bits
 import qualified Data.List as List
 import Data.Maybe (fromMaybe)
-import Data.Semigroup (Semigroup(stimes))
+import Data.Semigroup
+  (Semigroup(stimes), stimesIdempotent, stimesIdempotentMonoid)
 #if !(MIN_VERSION_base(4,11,0))
 import Data.Semigroup (Semigroup((<>)))
 #endif
-import Data.Semigroup (stimesIdempotentMonoid)
+#if MIN_VERSION_base(4,18,0)
+import qualified Data.Foldable1 as Foldable1
+import Data.List.NonEmpty (NonEmpty(..))
+#endif
 import Utils.Containers.Internal.Prelude hiding
   (filter, foldr, foldl, foldl', null, map)
 import Prelude ()
@@ -659,6 +667,40 @@ intersection (Tip kx1 bm1) t2 = intersectBM t2
 
 intersection Nil _ = Nil
 
+#if MIN_VERSION_base(4,18,0)
+-- | The intersection of a series of sets. Intersections are performed
+-- left-to-right.
+--
+-- @since FIXME
+intersections :: Foldable1.Foldable1 f => f IntSet -> IntSet
+intersections ss = case Foldable1.toNonEmpty ss of
+  s0 :| ss'
+    | null s0 -> empty
+    | otherwise -> List.foldr go id ss' s0
+  where
+    go s r acc
+      | null acc' = empty
+      | otherwise = r acc'
+      where
+        acc' = intersection acc s
+{-# INLINABLE intersections #-}
+#endif
+
+-- | @IntSet@s form a 'Semigroup' under 'intersection'.
+--
+-- A @Monoid@ instance is not defined because it would be impractical to
+-- construct @mempty@, the @IntSet@ containing all @Int@s.
+--
+-- @since FIXME
+newtype Intersection = Intersection { getIntersection :: IntSet }
+  deriving (Show, Eq, Ord)
+
+instance Semigroup Intersection where
+  Intersection s1 <> Intersection s2 = Intersection (intersection s1 s2)
+
+  stimes = stimesIdempotent
+  {-# INLINABLE stimes #-}
+
 {--------------------------------------------------------------------
   Symmetric difference
 --------------------------------------------------------------------}
diff --git a/containers/src/Data/Set.hs b/containers/src/Data/Set.hs
index ddfa9f02b..030b4e009 100644
--- a/containers/src/Data/Set.hs
+++ b/containers/src/Data/Set.hs
@@ -111,9 +111,13 @@ module Data.Set (
             , difference
             , (\\)
             , intersection
+#if MIN_VERSION_base(4,18,0)
+            , intersections
+#endif
             , symmetricDifference
             , cartesianProduct
             , disjointUnion
+            , Intersection(..)
 
             -- * Filter
             , S.filter
diff --git a/containers/src/Data/Set/Internal.hs b/containers/src/Data/Set/Internal.hs
index 805f5c361..9e310dda7 100644
--- a/containers/src/Data/Set/Internal.hs
+++ b/containers/src/Data/Set/Internal.hs
@@ -155,7 +155,9 @@ module Data.Set.Internal (
             , unions
             , difference
             , intersection
+#if MIN_VERSION_base(4,18,0)
             , intersections
+#endif
             , symmetricDifference
             , cartesianProduct
             , disjointUnion
@@ -240,7 +242,6 @@ import Control.Applicative (Const(..))
 import qualified Data.List as List
 import Data.Bits (shiftL, shiftR)
 import Data.Semigroup (Semigroup(stimes))
-import Data.List.NonEmpty (NonEmpty(..))
 #if !(MIN_VERSION_base(4,11,0))
 import Data.Semigroup (Semigroup((<>)))
 #endif
@@ -249,6 +250,10 @@ import Data.Functor.Classes
 import Data.Functor.Identity (Identity)
 import qualified Data.Foldable as Foldable
 import Control.DeepSeq (NFData(rnf))
+#if MIN_VERSION_base(4,18,0)
+import qualified Data.Foldable1 as Foldable1
+import Data.List.NonEmpty (NonEmpty(..))
+#endif
 
 import Utils.Containers.Internal.StrictPair
 import Utils.Containers.Internal.PtrEquality
@@ -894,21 +899,37 @@ intersection t1@(Bin _ x l1 r1) t2
 {-# INLINABLE intersection #-}
 #endif
 
--- | The intersection of a series of sets. Intersections are performed left-to-right.
-intersections :: Ord a => NonEmpty (Set a) -> Set a
-intersections (s0 :| ss) = List.foldr go id ss s0
-    where
-      go s r acc
-          | null acc = empty
-          | otherwise = r (intersection acc s)
+#if MIN_VERSION_base(4,18,0)
+-- | The intersection of a series of sets. Intersections are performed
+-- left-to-right.
+--
+-- @since FIXME
+intersections :: (Foldable1.Foldable1 f, Ord a) => f (Set a) -> Set a
+intersections ss = case Foldable1.toNonEmpty ss of
+  s0 :| ss'
+    | null s0 -> empty
+    | otherwise -> List.foldr go id ss' s0
+  where
+    go s r acc
+      | null acc' = empty
+      | otherwise = r acc'
+      where
+        acc' = intersection acc s
+{-# INLINABLE intersections #-}
+#endif
 
--- | Sets form a 'Semigroup' under 'intersection'.
+-- | @Set@s form a 'Semigroup' under 'intersection'.
+--
+-- @since FIXME
 newtype Intersection a = Intersection { getIntersection :: Set a }
     deriving (Show, Eq, Ord)
 
 instance (Ord a) => Semigroup (Intersection a) where
     (Intersection a) <> (Intersection b) = Intersection $ intersection a b
+    {-# INLINABLE (<>) #-}
+
     stimes = stimesIdempotent
+    {-# INLINABLE stimes #-}
 
 {--------------------------------------------------------------------
   Symmetric difference