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≤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 q2-1-r lines, then r=s(\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}+1) for an integer s≥2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}). We also discuss maximal partial spreads in PG(3,p3), p=p0h, p0 prime, p0 ≥ 5, h ≥ 1, p ≠ 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p2+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p3) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p3). In PG(3,p3),p square, for maximal partial spreads of deficiency δ ≤ p2+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 δ, the set of holes is a disjoint union of subgeometries PG(2t+1,\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}), which implies that δ ≡ 0 (mod\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}+1) and, when (2t+1)(\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}-1) <q-1, that δ ≥ 2(\documentclass[12pt]{minimal}
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$$\sqrt q$$
\end{document}+1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2,\documentclass[12pt]{minimal}
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$$\sqrt[3]{q}$$
\end{document}) and this implies δ ≡ 0 (mod q2/3+q1/3+1). A more general result is also presented.
