  In 1981 Edgar Enochs conjectured that every module has a flat cover and
finally proved this in 2001. Since then a great deal of effort has been spent
on studying different types of covers, for example injective and torsion free
covers. In 2008, Mahmoudi and Renshaw initiated the study of flat covers of
acts over monoids but their definition of cover was slightly different from
that of Enochs. Recently, Bailey and Renshaw produced some preliminary results
on the `other' type of cover and it is this work that is extended in this
paper. We consider free, divisible, torsion free and injective covers and
demonstrate that in some cases the results are quite different from the module
case.
