The 41-year-old Coleman will take charge of the national team that he played for 32 times. He quit as coach of Greek team Larissa on Jan. 9. He also has coached Fulham, Real Sociedad and Coventry.
Out of respect for Speed, who was found by his wife hanged at their home on Nov. 27, the Football Association of Wales waited to appoint his replacement.
Coleman will look to carry on the impressive work of his predecessor, who lifted the team from 116th to 48th in the FIFA rankings last year.
Wales hasn’t qualified for the finals of a major tournament since the 1958 World Cup.