(Updated - Thanks to the people who commented)

Modern Versions of PostgreSQL

Suppose you have a table named test1, to which you want to add an auto-incrementing, primary-key id (surrogate) column. The following command should be sufficient in recent versions of PostgreSQL:

   ALTER TABLE test1 ADD COLUMN id SERIAL PRIMARY KEY;

Older Versions of PostgreSQL

In old versions of PostgreSQL (prior to 8.x?) you had to do all the dirty work. The following sequence of commands should do the trick:

  ALTER TABLE test1 ADD COLUMN id INTEGER;  CREATE SEQUENCE test_id_seq OWNED BY test1.id;  ALTER TABLE test ALTER COLUMN id SET DEFAULT nextval('test_id_seq');  UPDATE test1 SET id = nextval('test_id_seq');

Again, in recent versions of Postgres this is roughly equivalent to the single command above.

ALTER TABLE test1 ADD COLUMN id SERIAL PRIMARY KEY;

This is all you need to:

1. Add the id column
2. Populate it with a sequence from 1 to count(*).
3. Set it as primary key / not null.

Credit is given to @resnyanskiy who gave this answer in a comment.

To use an identity column in v10,

ALTER TABLE test ADD COLUMN id { int | bigint | smallint}GENERATED { BY DEFAULT | ALWAYS } AS IDENTITY PRIMARY KEY;

For an explanation of identity columns, see https://blog.2ndquadrant.com/postgresql-10-identity-columns/.

For the difference between GENERATED BY DEFAULT and GENERATED ALWAYS, see https://www.cybertec-postgresql.com/en/sequences-gains-and-pitfalls/.

For altering the sequence, see https://popsql.io/learn-sql/postgresql/how-to-alter-sequence-in-postgresql/.