This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2r \leq q+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q^2+1-r lines, then r=s(\sqrt q+1) for an integer s\ge 2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,\sqrt{q}) . We also discuss maximal partial spreads in PG(3,p^3), p=p_0^h, p_0 prime, p_0 \geq 5, h \geq 1, p \neq 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p^2+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p^3) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p^3). In PG(3,p^3),p square, for maximal partial spreads of deficiency\delta \leq p^2+p+1 , the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies \delta, the set of holes is a disjoint union of subgeometries PG(2t+1,\sqrt{q}), which implies that \delta\equiv 0 \pmod{\sqrt{q}+1} and, when (2t+1)(\sqrt{q}-1) <q-1, that \delta \geq 2(\sqrt{q}+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,\sqrt[3]{q}) and this implies \delta \equiv 0 \pmod{q^{2/3}+q^{1/3}+1}. A more general result is also presented.