diff --git a/src/Magnum/Math/Quaternion.h b/src/Magnum/Math/Quaternion.h
index 838e6ecdd..a7274e744 100644
--- a/src/Magnum/Math/Quaternion.h
+++ b/src/Magnum/Math/Quaternion.h
@@ -39,6 +39,10 @@
 
 namespace Magnum { namespace Math {
 
+namespace Implementation {
+    template<class, class> struct QuaternionConverter;
+}
+
 /** @relatesalso Quaternion
 @brief Dot product between two quaternions
 
@@ -214,6 +218,14 @@ template<class T> class Quaternion {
          */
         constexpr explicit Quaternion(const Vector3<T>& vector): _vector(vector), _scalar(T(0)) {}
 
+        /** @brief Construct quaternion from external representation */
+        template<class U, class V = decltype(Implementation::QuaternionConverter<T, U>::from(std::declval<U>()))> constexpr explicit Quaternion(const U& other): Quaternion{Implementation::QuaternionConverter<T, U>::from(other)} {}
+
+        /** @brief Convert quaternion to external representation */
+        template<class U, class V = decltype(Implementation::QuaternionConverter<T, U>::to(std::declval<Quaternion<T>>()))> constexpr explicit operator U() const {
+            return Implementation::QuaternionConverter<T, U>::to(*this);
+        }
+
         /** @brief Equality comparison */
         bool operator==(const Quaternion<T>& other) const {
             return _vector == other._vector && TypeTraits<T>::equals(_scalar, other._scalar);
@@ -237,7 +249,7 @@ template<class T> class Quaternion {
         }
 
         /** @brief Vector part */
-        constexpr Vector3<T> vector() const { return _vector; }
+        constexpr const Vector3<T> vector() const { return _vector; }
 
         /** @brief Scalar part */
         constexpr T scalar() const { return _scalar; }
diff --git a/src/Magnum/Math/Test/QuaternionTest.cpp b/src/Magnum/Math/Test/QuaternionTest.cpp
index b9aba3231..115061eb1 100644
--- a/src/Magnum/Math/Test/QuaternionTest.cpp
+++ b/src/Magnum/Math/Test/QuaternionTest.cpp
@@ -29,7 +29,27 @@
 #include "Magnum/Math/Matrix4.h"
 #include "Magnum/Math/Quaternion.h"
 
-namespace Magnum { namespace Math { namespace Test {
+struct Quat {
+    float x, y, z, w;
+};
+
+namespace Magnum { namespace Math {
+
+namespace Implementation {
+
+template<> struct QuaternionConverter<Float, Quat> {
+    constexpr static Quaternion<Float> from(const Quat& other) {
+        return {{other.x, other.y, other.z}, other.w};
+    }
+
+    constexpr static Quat to(const Quaternion<Float>& other) {
+        return {other.vector().x(), other.vector().y(), other.vector().z(), other.scalar() };
+    }
+};
+
+}
+
+namespace Test {
 
 struct QuaternionTest: Corrade::TestSuite::Tester {
     explicit QuaternionTest();
@@ -38,6 +58,7 @@ struct QuaternionTest: Corrade::TestSuite::Tester {
     void constructDefault();
     void constructFromVector();
     void constructCopy();
+    void convert();
 
     void compare();
     void isNormalized();
@@ -80,6 +101,7 @@ QuaternionTest::QuaternionTest() {
               &QuaternionTest::constructDefault,
               &QuaternionTest::constructFromVector,
               &QuaternionTest::constructCopy,
+              &QuaternionTest::convert,
 
               &QuaternionTest::compare,
               &QuaternionTest::isNormalized,
@@ -139,6 +161,24 @@ void QuaternionTest::constructCopy() {
     CORRADE_COMPARE(b, Quaternion({1.0f, -3.0f, 7.0f}, 2.5f));
 }
 
+void QuaternionTest::convert() {
+    constexpr Quat a{1.5f, -3.5f, 7.0f, -0.5f};
+    constexpr Quaternion b{{1.5f, -3.5f, 7.0f}, -0.5f};
+
+    constexpr Quaternion c{a};
+    CORRADE_COMPARE(c, b);
+
+    constexpr Quat d(b);
+    CORRADE_COMPARE(d.x, a.x);
+    CORRADE_COMPARE(d.y, a.y);
+    CORRADE_COMPARE(d.z, a.z);
+    CORRADE_COMPARE(d.w, a.w);
+
+    /* Implicit conversion is not allowed */
+    CORRADE_VERIFY(!(std::is_convertible<Quat, Quaternion>::value));
+    CORRADE_VERIFY(!(std::is_convertible<Quaternion, Quat>::value));
+}
+
 void QuaternionTest::compare() {
     CORRADE_VERIFY(Quaternion({1.0f+TypeTraits<Float>::epsilon()/2, 2.0f, 3.0f}, -4.0f) == Quaternion({1.0f, 2.0f, 3.0f}, -4.0f));
     CORRADE_VERIFY(Quaternion({1.0f+TypeTraits<Float>::epsilon()*2, 2.0f, 3.0f}, -4.0f) != Quaternion({1.0f, 2.0f, 3.0f}, -4.0f));