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The paper contains a collection of results related to weight structures,$t$-structures, and (more generally) to torsion pairs. For any weight structure$w$ we study (co)homological pure functors; these "ignore all weights exceptweight zero" and have already found several applications. We also study virtual$t$-truncations of cohomological functors coming from $w$. These are closelyrelated to $t$-structures; so we prove in several cases (including certaincategories of coherent sheaves) that $w$ "gives" a $t$-structure (that isadjacent or $\Phi$-orthogonal to it). We also study in detail "well generated"weight structures (and prove that any perfect set of objects generates a weightstructure). The existence of weight structures right adjacent to compactlygenerated $t$-structures (and constructed using Brown-Comenetz duality) impliesthat the hearts of the latter have injective cogenerators and satisfy the AB3*axiom; actually, "most of them" are Grothendieck abelian (due to the existenceof "regularly orthogonal" weight structures).  It is convenient for us to use the notion of torsion pairs; these essentiallygeneralize both weight structures and $t$-structures. We prove severalproperties of torsion pairs (that are rather parallel to that of weightstructures); we also generalize a theorem of D. Pospisil and J. Stovicek toobtain a classification of compactly generated torsion pairs.

On torsion pairs, (well generated) weight structures, adjacent $t$-structures, and related (co)homological functors