diff --git a/src/Monolog/Handler/AbstractHandler.php b/src/Monolog/Handler/AbstractHandler.php
index 525e4f99..758a425c 100644
--- a/src/Monolog/Handler/AbstractHandler.php
+++ b/src/Monolog/Handler/AbstractHandler.php
@@ -169,6 +169,8 @@ abstract class AbstractHandler implements HandlerInterface
             $this->close();
         } catch (\Exception $e) {
             // do nothing
+        } catch (\Throwable $e) {
+            // do nothing
         }
     }
 
diff --git a/src/Monolog/Handler/WhatFailureGroupHandler.php b/src/Monolog/Handler/WhatFailureGroupHandler.php
index 05a88173..2732ba3d 100644
--- a/src/Monolog/Handler/WhatFailureGroupHandler.php
+++ b/src/Monolog/Handler/WhatFailureGroupHandler.php
@@ -35,6 +35,8 @@ class WhatFailureGroupHandler extends GroupHandler
                 $handler->handle($record);
             } catch (\Exception $e) {
                 // What failure?
+            } catch (\Throwable $e) {
+                // What failure?
             }
         }
 
@@ -51,6 +53,8 @@ class WhatFailureGroupHandler extends GroupHandler
                 $handler->handleBatch($records);
             } catch (\Exception $e) {
                 // What failure?
+            } catch (\Throwable $e) {
+                // What failure?
             }
         }
     }