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We introduce a generalization of tilting modules of finite projectivedimension, projectively Wakamatsu tilting modules, which are self-orthogonaland Ext-progenerators in their Ext-perpendicular categories. Under a certainfiniteness condition, we prove that the following modules coincide:projectively Wakamatsu tilting, Wakamatsu tilting, maximal self-orthogonal, andself-orthogonal modules with the same rank as the algebra. This providesanother proof of the weak Gorensteinness of representation-finite algebras. Toprove this, we introduce Bongartz completion of self-orthogonal modules andcharacterize its existence. Moreover, we study a binary relation on Wakamatsutilting modules which extends the poset of tilting modules, and use it to provethat every self-orthogonal module over a representation-finiteIwanaga-Gorenstein algebra has finite projective dimension. Finally, we discussseveral conjectures related to self-orthogonal modules and their connections tofamous homological conjectures.

Maximal self-orthogonal modules and a new generalization of tilting modules