  The notion of a Heffter array, which received much attention in the last
decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we
study the existence problem of a set of $r$ mutually orthogonal Heffter systems
for any $r$. Such a set is equivalent to a resolvable partial linear space of
degree $r$ whose parallel classes are Heffter systems: this is a new
combinatorial design that we call a Heffter space. We present a series of
direct constructions of Heffter spaces with block size odd and arbitrarily
large degree $r$ obtained with the crucial use of finite fields. Among the
applications we establish, in particular, the existence of $r$ mutually
orthogonal $k$-cycle systems of order a prime power $q=2kw+1$ whenever $kw$ is
odd and $w>4k^4\lceil{r\over k}\rceil$.
