A review engagement
In a review engagement the accountant is engaged to review but not audit financial or other information.
The work performed is less detailed than for an audit. As a result, the level of assurance provided is seen to be limited as opposed to the high assurance given by an audit.
A review engagement could be:

• an attestation engagement, or
• a direct reporting engagement.
  In an attestation engagement, an accountant might be engaged to attest to (vouch for) the fact that certain procedures within an entity have been performed. He will not comment on the quality of the procedures, merely that they have been performed.
  In a direct reporting engagement, an accountant will carry out an independent examination of financial or other information prepared by his client for use by a third party. The resultant special report will be on some aspect of his client’s affairs and will usually provide negative assurance.
  One of the most common forms of direct reporting engagement is ‘due diligence’ work. In the context of mergers and takeovers, Entity X, taking over Entity Y, is likely to require a due diligence report into the valuation of the assets and liabilities of Entity Y.

Agreed-upon procedures
In an agreed-upon procedures engagement, the client, as opposed to the accountant, determines the procedures to be followed.
The accountant provides a report on factual findings from the procedures performed. No assurance is provided.
It is left to the user to assess the procedures and findings and to draw their own conclusions.

A compilation engagement
In a compilation engagement the accountant is engaged to prepare (but not review or audit) financial statements or other information. A typical example is the preparation of financial statements for a sole trader or a club.
The accountant is engaged for his accounting, not auditing, expertise. No assurance is provided. The work normally involves the collection and summarising of information.
A ‘compilation report’ may be issued at the end of the assignment.