Content: PrefaceZF theory and some point sets on the real line Countable versions of AC and real analysis Uncountable versions of AC and Lebesgue nonmeasurable sets The Continuum Hypothesis and Lebesgue nonmeasurable sets Measurability properties of sets and functions Radon measures and nonmeasurable sets Real-valued step functions with strange measurability properties A partition of the real line into continuum many thick subsets Measurability properties of Vitali sets A relationship between the measurability and continuity of real-valued functions A relationship between absolutely nonmeasurable functions and Sierpi'nski-Zygmund type functions Sums of absolutely nonmeasurable injective functions A large group of absolutely nonmeasurable additive functions Additive properties of certain classes of pathological functions Absolutely nonmeasurable homomorphisms of commutative groups Measurable and nonmeasurable sets with homogeneous sections A combinatorial problem on translation invariant extensions of the Lebesgue measure Countable almost invariant partitions of G-spaces Nonmeasurable unions of measure zero sections of plane sets Measurability properties of well-orderings Appendix 1: The axioms of set theory Appendix 2: The Axiom of Choice and Generalized Continuum Hypothesis Appendix 3: Martin's Axiom and its consequences in real analysis Appendix 4: !1-dense subsets of the real line Appendix 5: The beginnings of descriptive set theory Bibliography Subject Index