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Introductory lecture summary: 1. Finite dimensional hilbert spaces and (complemented) modular lattices; infinite-dimensional hilbert spaces and orthomodularity. 2. von Neumann's QL; von Neumann-Birkhoff (briefly!); reconstruction of QM from P(H) 3. Mackey's programme; some early axiomatics (e.g., Zierler); QLs as OMPs + order-determining sets of states 4. Piron's Theorem; some discussion of Piron's axioms 5. Keller's examples (maybe just a mention, though I'd like to indicate how they come up); Soler's theorem (just the statement) 6. Abstract OMLs; Greechie Diagrams and the Loop Lemma; brief mention of Harding's results on decompositions 7. Tensor products (the F-R example, showing OMLs not stable under tensor products); orthoalgebras. 8. Orthoalgebras from test spaces