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Finding the current session's ID

select sid, serial# from v$session where audsid = sys_context('userenv','sessionid');
Or, if you're only interested in the sid, not in serial#: 
select distinct sid from v$mystat;
Note, the selected sid has nothing to do with the System Identifier.
If dbms_support is installed, the current sid can also be found with its mysid function: 
select dbms_support.mysid from dual;
Thanks to Jan Arnoldus who pointed out an error on this page!