Adding variables to make scores from questionnaire data in SPSS

Very often students who have questionnaire data ask me whether they can add together some similar variables to create a new variable that summarizes the meaning of all the questions that go into it, for example if you have a load of questions about the service of an airline, the sum of those questions gives a new variable that may be interpreted as airline satisfaction score. The answers is YES (that new variable is called a score/index), but there are things to watch out for.
Keywords: index, composite score, attribute, trait, items, unit weighted scores, weighted scores, reliability, MEAN().