A finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$
F
q
equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-$$(n,k,\lambda )$$
(
n
,
k
,
λ
)
design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly $$\lambda $$
λ
members of Y. Nontrivial examples are currently only known for $$t\le 2$$
t
≤
2
. We show that t-$$(n,k,\lambda )$$
(
n
,
k
,
λ
)
designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$
k
>
21
2
t
and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.