Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4)\ PG(3, 2) with r lines, where r {6, 7}, to construct transversal-free translation nets of order 16 and degree r +3 and hence maximal sets of r +1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q+(2m +1, q), and yields infinite classes of q2n-1 -1 MAXMOLS(q2n), for n 2 and q a power of two, and for n = 2 and q a power of three.