diff --git a/doc/src/sgml/ref/reindex.sgml b/doc/src/sgml/ref/reindex.sgml
index 6750eb6e47..fcbda88149 100644
--- a/doc/src/sgml/ref/reindex.sgml
+++ b/doc/src/sgml/ref/reindex.sgml
@@ -22,8 +22,7 @@ PostgreSQL documentation
  <refsynopsisdiv>
 <synopsis>
 REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] { INDEX | TABLE | SCHEMA } [ CONCURRENTLY ] <replaceable class="parameter">name</replaceable>
-REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] DATABASE [ CONCURRENTLY ] [ <replaceable class="parameter">name</replaceable> ]
-REINDEX [ ( <replaceable class="parameter">option</replaceable> [, ...] ) ] SYSTEM [ <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>
 
diff --git a/src/backend/parser/gram.y b/src/backend/parser/gram.y
index ace4fb5c77..f9037761f9 100644
--- a/src/backend/parser/gram.y
+++ b/src/backend/parser/gram.y
@@ -564,7 +564,7 @@ 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
+%type <ival>	reindex_target_relation reindex_target_all
 %type <list>	opt_reindex_option_list
 
 %type <node>	copy_generic_opt_arg copy_generic_opt_arg_list_item
@@ -9092,13 +9092,12 @@ DropTransformStmt: DROP TRANSFORM opt_if_exists FOR Typename LANGUAGE name opt_d
  *
  *		QUERY:
  *
- *		REINDEX [ (options) ] {TABLE | INDEX | SCHEMA} [CONCURRENTLY] <name>
- *		REINDEX [ (options) ] DATABASE [CONCURRENTLY] [<name>]
- *		REINDEX [ (options) ] SYSTEM [<name>]
+ *		REINDEX [ (options) ] {INDEX | TABLE | SCHEMA} [CONCURRENTLY] <name>
+ *		REINDEX [ (options) ] {DATABASE | SYSTEM} [CONCURRENTLY] [<name>]
  *****************************************************************************/
 
 ReindexStmt:
-			REINDEX opt_reindex_option_list reindex_target_type opt_concurrently qualified_name
+			REINDEX opt_reindex_option_list reindex_target_relation opt_concurrently qualified_name
 				{
 					ReindexStmt *n = makeNode(ReindexStmt);
 
@@ -9116,37 +9115,36 @@ ReindexStmt:
 					ReindexStmt *n = makeNode(ReindexStmt);
 
 					n->kind = REINDEX_OBJECT_SCHEMA;
-					n->name = $5;
 					n->relation = NULL;
+					n->name = $5;
 					n->params = $2;
 					if ($4)
 						n->params = lappend(n->params,
 											makeDefElem("concurrently", NULL, @4));
 					$$ = (Node *) n;
 				}
-			| REINDEX opt_reindex_option_list DATABASE opt_concurrently opt_single_name
+			| REINDEX opt_reindex_option_list reindex_target_all opt_concurrently opt_single_name
 				{
 					ReindexStmt *n = makeNode(ReindexStmt);
-					n->kind = REINDEX_OBJECT_DATABASE;
-					n->name = NULL;
-					n->relation = NULL;
-					n->params = $2;
-					$$ = (Node *) n;
-				}
-			| REINDEX opt_reindex_option_list SYSTEM_P opt_single_name
-				{
-					ReindexStmt *n = makeNode(ReindexStmt);
-					n->kind = REINDEX_OBJECT_SYSTEM;
-					n->name = NULL;
+
+					n->kind = $3;
 					n->relation = NULL;
+					n->name = $5;
 					n->params = $2;
+					if ($4)
+						n->params = lappend(n->params,
+											makeDefElem("concurrently", NULL, @4));
 					$$ = (Node *) n;
 				}
 		;
-reindex_target_type:
+reindex_target_relation:
 			INDEX					{ $$ = REINDEX_OBJECT_INDEX; }
 			| TABLE					{ $$ = REINDEX_OBJECT_TABLE; }
 		;
+reindex_target_all:
+			SYSTEM_P				{ $$ = REINDEX_OBJECT_SYSTEM; }
+			| DATABASE				{ $$ = REINDEX_OBJECT_DATABASE; }
+		;
 opt_reindex_option_list:
 			'(' utility_option_list ')'				{ $$ = $2; }
 			| /* EMPTY */							{ $$ = NULL; }
diff --git a/src/bin/scripts/t/090_reindexdb.pl b/src/bin/scripts/t/090_reindexdb.pl
index b5fff5a9cf..e706d686e3 100644
--- a/src/bin/scripts/t/090_reindexdb.pl
+++ b/src/bin/scripts/t/090_reindexdb.pl
@@ -40,12 +40,12 @@ my $toast_index = $node->safe_psql('postgres',
 # REINDEX operations.  A set of relfilenodes is saved from the catalogs
 # and then compared with pg_class.
 $node->safe_psql('postgres',
-	'CREATE TABLE index_relfilenodes (parent regclass, indname regclass, relfilenode oid);'
+	'CREATE TABLE index_relfilenodes (parent regclass, indname text, indoid oid, relfilenode oid);'
 );
 # Save the relfilenode of a set of toast indexes, one from the catalog
 # pg_constraint and one from the test table.
 my $fetch_toast_relfilenodes =
-  qq{SELECT b.oid::regclass, c.oid::regclass, c.relfilenode
+  qq{SELECT b.oid::regclass, c.oid::regclass::text, c.oid, c.relfilenode
   FROM pg_class a
     JOIN pg_class b ON (a.oid = b.reltoastrelid)
     JOIN pg_index i on (a.oid = i.indrelid)
@@ -53,7 +53,7 @@ my $fetch_toast_relfilenodes =
   WHERE b.oid IN ('pg_constraint'::regclass, 'test1'::regclass)};
 # Same for relfilenodes of normal indexes.  This saves the relfilenode
 # from an index of pg_constraint, and from the index of the test table.
-my $fetch_index_relfilenodes = qq{SELECT i.indrelid, a.oid, a.relfilenode
+my $fetch_index_relfilenodes = qq{SELECT i.indrelid, a.oid::regclass::text, a.oid, a.relfilenode
   FROM pg_class a
     JOIN pg_index i ON (i.indexrelid = a.oid)
   WHERE a.relname IN ('pg_constraint_oid_index', 'test1x')};
@@ -69,6 +69,8 @@ my $save_relfilenodes =
 # parent table is included to provide more context.
 my $compare_relfilenodes = qq(SELECT b.parent::regclass,
   regexp_replace(b.indname::text, '(pg_toast.pg_toast_)\\d+(_index)', '\\1<oid>\\2'),
+  CASE WHEN a.oid = b.indoid THEN 'OID is unchanged'
+    ELSE 'OID has changed' END,
   CASE WHEN a.relfilenode = b.relfilenode THEN 'relfilenode is unchanged'
     ELSE 'relfilenode has changed' END
   FROM index_relfilenodes b
@@ -83,10 +85,10 @@ $node->issues_sql_like(
 	'SQL REINDEX run');
 my $relnode_info = $node->safe_psql('postgres', $compare_relfilenodes);
 is( $relnode_info,
-	qq(pg_constraint|pg_constraint_oid_index|relfilenode is unchanged
-pg_constraint|pg_toast.pg_toast_<oid>_index|relfilenode is unchanged
-test1|pg_toast.pg_toast_<oid>_index|relfilenode has changed
-test1|test1x|relfilenode has changed),
+	qq(pg_constraint|pg_constraint_oid_index|OID is unchanged|relfilenode is unchanged
+pg_constraint|pg_toast.pg_toast_<oid>_index|OID is unchanged|relfilenode is unchanged
+test1|pg_toast.pg_toast_<oid>_index|OID is unchanged|relfilenode has changed
+test1|test1x|OID is unchanged|relfilenode has changed),
 	'relfilenode change after REINDEX DATABASE');
 
 # Re-save and run the second one.
@@ -98,10 +100,10 @@ $node->issues_sql_like(
 	'reindex system tables');
 $relnode_info = $node->safe_psql('postgres', $compare_relfilenodes);
 is( $relnode_info,
-	qq(pg_constraint|pg_constraint_oid_index|relfilenode has changed
-pg_constraint|pg_toast.pg_toast_<oid>_index|relfilenode has changed
-test1|pg_toast.pg_toast_<oid>_index|relfilenode is unchanged
-test1|test1x|relfilenode is unchanged),
+	qq(pg_constraint|pg_constraint_oid_index|OID is unchanged|relfilenode has changed
+pg_constraint|pg_toast.pg_toast_<oid>_index|OID is unchanged|relfilenode has changed
+test1|pg_toast.pg_toast_<oid>_index|OID is unchanged|relfilenode is unchanged
+test1|test1x|OID is unchanged|relfilenode is unchanged),
 	'relfilenode change after REINDEX SYSTEM');
 
 $node->issues_sql_like(
@@ -132,11 +134,22 @@ $node->issues_sql_like(
 	qr/statement: REINDEX \(VERBOSE, TABLESPACE $tbspace_name\) TABLE public\.test1;/,
 	'reindex with verbose output and tablespace');
 
-# the same with --concurrently
+# Same with --concurrently.
+# Save the state of the relations and compare them after the DATABASE
+# rebuild.
+$node->safe_psql('postgres',
+	"TRUNCATE index_relfilenodes; $save_relfilenodes");
 $node->issues_sql_like(
 	[ 'reindexdb', '--concurrently', 'postgres' ],
 	qr/statement: REINDEX DATABASE CONCURRENTLY postgres;/,
 	'SQL REINDEX CONCURRENTLY run');
+$relnode_info = $node->safe_psql('postgres', $compare_relfilenodes);
+is( $relnode_info,
+	qq(pg_constraint|pg_constraint_oid_index|OID is unchanged|relfilenode is unchanged
+pg_constraint|pg_toast.pg_toast_<oid>_index|OID is unchanged|relfilenode is unchanged
+test1|pg_toast.pg_toast_<oid>_index|OID has changed|relfilenode has changed
+test1|test1x|OID has changed|relfilenode has changed),
+	'OID change after REINDEX DATABASE CONCURRENTLY');
 
 $node->issues_sql_like(
 	[ 'reindexdb', '--concurrently', '-t', 'test1', 'postgres' ],
diff --git a/src/test/regress/expected/create_index.out b/src/test/regress/expected/create_index.out
index a913a1846f..6cd57e3eaa 100644
--- a/src/test/regress/expected/create_index.out
+++ b/src/test/regress/expected/create_index.out
@@ -2521,9 +2521,7 @@ ERROR:  cannot reindex system catalogs concurrently
 REINDEX INDEX CONCURRENTLY pg_toast.pg_toast_1260_index; -- no catalog toast index
 ERROR:  cannot reindex system catalogs concurrently
 REINDEX SYSTEM CONCURRENTLY postgres; -- not allowed for SYSTEM
-ERROR:  syntax error at or near "CONCURRENTLY"
-LINE 1: REINDEX SYSTEM CONCURRENTLY postgres;
-                       ^
+ERROR:  cannot reindex system catalogs concurrently
 REINDEX (CONCURRENTLY) SYSTEM postgres; -- ditto
 ERROR:  cannot reindex system catalogs concurrently
 REINDEX (CONCURRENTLY) SYSTEM;  -- ditto
@@ -2531,6 +2529,9 @@ ERROR:  cannot reindex system catalogs concurrently
 -- Warns about catalog relations
 REINDEX SCHEMA CONCURRENTLY pg_catalog;
 WARNING:  cannot reindex system catalogs concurrently, skipping all
+-- Not the current database
+REINDEX DATABASE not_current_database;
+ERROR:  can only reindex the currently open database
 -- Check the relation status, there should not be invalid indexes
 \d concur_reindex_tab
          Table "public.concur_reindex_tab"
diff --git a/src/test/regress/sql/create_index.sql b/src/test/regress/sql/create_index.sql
index 4b75790e47..a3738833b2 100644
--- a/src/test/regress/sql/create_index.sql
+++ b/src/test/regress/sql/create_index.sql
@@ -1076,6 +1076,8 @@ REINDEX (CONCURRENTLY) SYSTEM postgres; -- ditto
 REINDEX (CONCURRENTLY) SYSTEM;  -- ditto
 -- Warns about catalog relations
 REINDEX SCHEMA CONCURRENTLY pg_catalog;
+-- Not the current database
+REINDEX DATABASE not_current_database;
 
 -- Check the relation status, there should not be invalid indexes
 \d concur_reindex_tab