Pseudo-quadratic forms have been introduced by Tits in his Buildings of spherical type and finite BN-pairs (1974), in view of the classification of polar spaces. A slightly different notion is proposed by Tits and Weiss. In this paper we propose a generalization. With its help we will be able to clarify a few points in the classification of embedded polar spaces. We recall that, according to Tits’ book, given a division ring K and an admissible pair (σ,e) in it, the codomain of a (σ,e)-quadratic form is the group K¯:=K∕Kσ,e, where Kσ,e:={t−tσe}t∈K. Our generalization amounts to replace K¯ with a quotient K¯∕R¯ for a subgroup R¯ of K¯ such that λσR¯λ=R¯ for any λ∈K. We call generalized pseudo-quadratic forms (also generalized (σ,e)-quadratic forms) the forms defined in this more general way, keeping the words pseudo-quadratic form and (σ,e)-quadratic form for those defined as in Tits’ book. Generalized pseudo-quadratic forms behave just like pseudo-quadratic forms. In particular, every non-trivial generalized pseudo-quadratic form admits a unique sesquilinearization, characterized by the same property as the sesquilinearization of a pseudo-quadratic form. Moreover, if q:V→K¯∕R¯ is a non-trivial generalized pseudo-quadratic form and f:V×V→K is its sesquilinearization, the points and the lines of PG(V) where q vanishes form a subspace Sq of the polar space Sf associated to f. In this paper, after a discussion of quotients and covers of generalized pseudo-quadratic forms, we shall prove the following, which sharpens a celebretated theorem of Buekenhout and Lefevre. Let e:S→ PG(V) be a projective embedding of a non-degenerate polar space S of rank at least 2; then e(S) is either the polar space Sq associated to a generalized pseudo-quadratic form q or the polar space Sf associated to an alternating form f. By exploiting this theorem we also obtain an elementary proof of the following well known fact: an embedding e as above is dominant if and only if either e(S)=Sq for a pseudo-quadratic form q or char(K)≠2 and e(S)=Sf for an alternating form f.