In this chapter we summarize some homological preliminaries needed for our investigations in the sequel. In particular, we investigate unbounded complexes. To this end some additional considerations for their resolutions are necessary, that is, we report and summarize part of the work of Avramov and Foxby resp. Spaltenstein (see Avramov, Foxby (J Pure Appl Algebra, 71, 129–155, (1991), ), Spaltenstein (Compos Math, 65, 121–154, (1988), )) not available in this form elsewhere. After recalling some results about double complexes we start with the extension to complexes of results on the microscope and telescope introduced by Greenlees and May in Greenlees, May (J Algebra, 149, 438–453, (1992), ). This is used in order to get certain resolutions of unbounded complexes, as suggested by Avramov and Foxby. We also discuss minimal injective resolutions for unbounded complexes.