  We study the Mathias--Prikry and the Laver type forcings associated with
filters and coideals. We isolate a crucial combinatorial property of Mathias
reals, and prove that Mathias--Prikry forcings with summable ideals are all
mutually bi-embeddable. We show that Mathias forcing associated with the
complement of an analytic ideal always adds a dominating real. We also
characterize filters for which the associated Mathias--Prikry forcing does not
add eventually different reals, and show that they are countably generated
provided they are Borel. We give a characterization of $\omega$-hitting and
$\omega$-splitting families which retain their property in the extension by a
Laver type forcing associated with a coideal.
