diff --git a/doc/src/sgml/ref/reindex.sgml b/doc/src/sgml/ref/reindex.sgml
index 336ca24b31..143847dfa9 100644
--- a/doc/src/sgml/ref/reindex.sgml
+++ b/doc/src/sgml/ref/reindex.sgml
@@ -21,7 +21,8 @@ PostgreSQL documentation
 
  <refsynopsisdiv>
 <synopsis>
-REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { INDEX | TABLE | SCHEMA | DATABASE | SYSTEM } [ CONCURRENTLY ] <replaceable class="parameter">name</replaceable>
+REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { INDEX | TABLE | SCHEMA } [ CONCURRENTLY ] <replaceable class="parameter">name</replaceable>
+REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { DATABASE | SYSTEM } [ CONCURRENTLY ] [ <replaceable class="parameter">name</replaceable> ]
 
 <phrase>where <replaceable class="parameter">option</replaceable> can be one of:</phrase>
 
@@ -126,8 +127,9 @@ REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { IN
     <term><literal>DATABASE</literal></term>
     <listitem>
      <para>
-      Recreate all indexes within the current database.
-      Indexes on shared system catalogs are also processed.
+      Recreate all indexes within the current database, except system
+      catalogs.
+      Indexes on shared system catalogs are not processed.
       This form of <command>REINDEX</command> cannot be executed inside a
       transaction block.
      </para>
@@ -155,7 +157,7 @@ REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { IN
       reindexed.  Index and table names can be schema-qualified.
       Presently, <command>REINDEX DATABASE</command> and <command>REINDEX SYSTEM</command>
       can only reindex the current database, so their parameter must match
-      the current database's name.
+      the current database's name, if supplied, though supplying the name is optional.
      </para>
     </listitem>
    </varlistentry>
diff --git a/src/backend/commands/indexcmds.c b/src/backend/commands/indexcmds.c
index 99f5ab83c3..df10262f73 100644
--- a/src/backend/commands/indexcmds.c
+++ b/src/backend/commands/indexcmds.c
@@ -2877,10 +2877,13 @@ ReindexMultipleTables(const char *objectName, ReindexObjectType objectKind,
 	bool		concurrent_warning = false;
 	bool		tablespace_warning = false;
 
-	AssertArg(objectName);
 	Assert(objectKind == REINDEX_OBJECT_SCHEMA ||
 		   objectKind == REINDEX_OBJECT_SYSTEM ||
 		   objectKind == REINDEX_OBJECT_DATABASE);
+	/*
+	 * This matches the options enforced by the grammar.
+	 */
+	AssertArg(objectName || objectKind != REINDEX_OBJECT_SCHEMA);
 
 	if (objectKind == REINDEX_OBJECT_SYSTEM &&
 		(params->options & REINDEXOPT_CONCURRENTLY) != 0)
@@ -2906,13 +2909,13 @@ ReindexMultipleTables(const char *objectName, ReindexObjectType objectKind,
 	{
 		objectOid = MyDatabaseId;
 
-		if (strcmp(objectName, get_database_name(objectOid)) != 0)
+		if (objectName && strcmp(objectName, get_database_name(objectOid)) != 0)
 			ereport(ERROR,
 					(errcode(ERRCODE_FEATURE_NOT_SUPPORTED),
 					 errmsg("can only reindex the currently open database")));
 		if (!pg_database_ownercheck(objectOid, GetUserId()))
 			aclcheck_error(ACLCHECK_NOT_OWNER, OBJECT_DATABASE,
-						   objectName);
+						   get_database_name(objectOid));
 	}
 
 	/*
@@ -2970,10 +2973,24 @@ ReindexMultipleTables(const char *objectName, ReindexObjectType objectKind,
 			!isTempNamespace(classtuple->relnamespace))
 			continue;
 
-		/* Check user/system classification, and optionally skip */
-		if (objectKind == REINDEX_OBJECT_SYSTEM &&
-			!IsSystemClass(relid, classtuple))
-			continue;
+		/*
+		 * Check user/system classification, and skip appropriately.
+		 */
+		if (objectKind == REINDEX_OBJECT_SYSTEM)
+		{
+			if (!IsSystemClass(relid, classtuple))
+				continue;
+		}
+		else if (objectKind == REINDEX_OBJECT_DATABASE)
+		{
+			/*
+			 * You might think it would be good to include catalogs,
+			 * but doing that can deadlock, so isn't much use in real world,
+			 * nor can we safely test that it even works.
+			 */
+			if (IsSystemClass(relid, classtuple))
+				continue;
+		}
 
 		/*
 		 * The table can be reindexed if the user is superuser, the table
diff --git a/src/backend/parser/gram.y b/src/backend/parser/gram.y
index 969c9c158f..25a5192595 100644
--- a/src/backend/parser/gram.y
+++ b/src/backend/parser/gram.y
@@ -560,7 +560,7 @@ static Node *makeRecursiveViewSelect(char *relname, List *aliases, Node *query);
 %type <defelt>	generic_option_elem alter_generic_option_elem
 %type <list>	generic_option_list alter_generic_option_list
 
-%type <ival>	reindex_target_type reindex_target_multitable
+%type <ival>	reindex_target_type reindex_target_multitable reindex_name_optional
 
 %type <node>	copy_generic_opt_arg copy_generic_opt_arg_list_item
 %type <defelt>	copy_generic_opt_elem
@@ -9105,6 +9105,24 @@ ReindexStmt:
 											makeDefElem("concurrently", NULL, @3));
 					$$ = (Node *) n;
 				}
+			| REINDEX reindex_name_optional
+				{
+					ReindexStmt *n = makeNode(ReindexStmt);
+					n->kind = $2;
+					n->name = NULL;
+					n->relation = NULL;
+					n->params = NIL;
+					$$ = (Node *)n;
+				}
+			| REINDEX '(' utility_option_list ')' reindex_name_optional
+				{
+					ReindexStmt *n = makeNode(ReindexStmt);
+					n->kind = $5;
+					n->name = NULL;
+					n->relation = NULL;
+					n->params = $3;
+					$$ = (Node *)n;
+				}
 			| REINDEX '(' utility_option_list ')' reindex_target_type opt_concurrently qualified_name
 				{
 					ReindexStmt *n = makeNode(ReindexStmt);
@@ -9141,6 +9159,11 @@ reindex_target_multitable:
 			| SYSTEM_P				{ $$ = REINDEX_OBJECT_SYSTEM; }
 			| DATABASE				{ $$ = REINDEX_OBJECT_DATABASE; }
 		;
+/* For these options the name is optional */
+reindex_name_optional:
+			SYSTEM_P				{ $$ = REINDEX_OBJECT_SYSTEM; }
+			| DATABASE				{ $$ = REINDEX_OBJECT_DATABASE; }
+		;
 
 /*****************************************************************************
  *
diff --git a/src/bin/scripts/reindexdb.c b/src/bin/scripts/reindexdb.c
index f3b03ec325..0e93a4eeff 100644
--- a/src/bin/scripts/reindexdb.c
+++ b/src/bin/scripts/reindexdb.c
@@ -360,18 +360,6 @@ reindex_one_database(ConnParams *cparams, ReindexType type,
 		{
 			case REINDEX_DATABASE:
 
-				/*
-				 * Database-wide parallel reindex requires special processing.
-				 * If multiple jobs were asked, we have to reindex system
-				 * catalogs first as they cannot be processed in parallel.
-				 */
-				if (concurrently)
-					pg_log_warning("cannot reindex system catalogs concurrently, skipping all");
-				else
-					run_reindex_command(conn, REINDEX_SYSTEM, PQdb(conn), echo,
-										verbose, concurrently, false,
-										tablespace);
-
 				/* Build a list of relations from the database */
 				process_list = get_parallel_object_list(conn, process_type,
 														user_list, echo);
diff --git a/src/bin/scripts/t/090_reindexdb.pl b/src/bin/scripts/t/090_reindexdb.pl
index 398fc4e6bb..fd422b3d3d 100644
--- a/src/bin/scripts/t/090_reindexdb.pl
+++ b/src/bin/scripts/t/090_reindexdb.pl
@@ -176,11 +176,6 @@ $node->command_fails(
 $node->command_fails(
 	[ 'reindexdb', '-j', '2', '-i', 'i1', 'postgres' ],
 	'parallel reindexdb cannot process indexes');
-$node->issues_sql_like(
-	[ 'reindexdb', '-j', '2', 'postgres' ],
-	qr/statement:\ REINDEX SYSTEM postgres;
-.*statement:\ REINDEX TABLE public\.test1/s,
-	'parallel reindexdb for database issues REINDEX SYSTEM first');
 # Note that the ordering of the commands is not stable, so the second
 # command for s2.t2 is not checked after.
 $node->issues_sql_like(
@@ -190,13 +185,8 @@ $node->issues_sql_like(
 $node->command_ok(
 	[ 'reindexdb', '-j', '2', '-S', 's3' ],
 	'parallel reindexdb with empty schema');
-$node->command_checks_all(
+$node->command_ok(
 	[ 'reindexdb', '-j', '2', '--concurrently', '-d', 'postgres' ],
-	0,
-	[qr/^$/],
-	[
-		qr/^reindexdb: warning: cannot reindex system catalogs concurrently, skipping all/s
-	],
-	'parallel reindexdb for system with --concurrently skips catalogs');
+	'parallel reindexdb with empty schema, concurrently');
 
 done_testing();
diff --git a/src/test/regress/expected/create_index.out b/src/test/regress/expected/create_index.out
index d55aec3a1d..e697da6e81 100644
--- a/src/test/regress/expected/create_index.out
+++ b/src/test/regress/expected/create_index.out
@@ -2808,6 +2808,12 @@ ERROR:  REINDEX SCHEMA cannot run inside a transaction block
 END;
 -- concurrently
 REINDEX SCHEMA CONCURRENTLY schema_to_reindex;
+-- unqualified reindex database
+-- if you want to test REINDEX DATABASE, uncomment the following line,
+-- but note that this adds about 0.5s to the regression tests and the
+-- results are volatile when run in parallel to other tasks. Note also
+-- that REINDEX SYSTEM is specifically not tested because it can deadlock.
+-- REINDEX (VERBOSE) DATABASE;
 -- Failure for unauthorized user
 CREATE ROLE regress_reindexuser NOLOGIN;
 SET SESSION ROLE regress_reindexuser;
diff --git a/src/test/regress/sql/create_index.sql b/src/test/regress/sql/create_index.sql
index d8fded3d93..19abc1d210 100644
--- a/src/test/regress/sql/create_index.sql
+++ b/src/test/regress/sql/create_index.sql
@@ -1235,6 +1235,13 @@ END;
 -- concurrently
 REINDEX SCHEMA CONCURRENTLY schema_to_reindex;
 
+-- unqualified reindex database
+-- if you want to test REINDEX DATABASE, uncomment the following line,
+-- but note that this adds about 0.5s to the regression tests and the
+-- results are volatile when run in parallel to other tasks. Note also
+-- that REINDEX SYSTEM is specifically not tested because it can deadlock.
+-- REINDEX (VERBOSE) DATABASE;
+
 -- Failure for unauthorized user
 CREATE ROLE regress_reindexuser NOLOGIN;
 SET SESSION ROLE regress_reindexuser;