Probability theory is usually based on a probability space \((\Omega, {\mathcal A},P)\) where the sets \(A\in{\mathcal A}\), resp., the indicator functions \(I_A\), are interpreted as events and the measurable maps \(\widehat X: \Omega\to \mathbb{R}\), i.e., the random variables, as observable quantities. Fuzzy probability theory, as understood by the author and some others [\textit{S. Bugajski}, Int. J. Theor. Phys. 35, No. 11, 2229-2244 (1996; Zbl 0872.60003); \textit{S. Bugajski}, \textit{K.-E. Hellwig} and \textit{W. Stulpe}, Rep. Math. Phys. 41, No. 1, 1-11 (1998)], is a generalization of standard probability theory where so-called fuzzy events are described by arbitrary measurable functions \(f:\Omega \to[0,1]\) and the integral \(\int fdP\) is interpreted to be the probability for the occurrence of \(f\). Denoting the set of all fuzzy events \(f\) by \({\mathcal E}(\Omega, {\mathcal A})\) and the Borel sets of the real line by \({\mathcal B}(\mathbb{R})\), a fuzzy random variable, i.e., an observable quantity, is given by a map \(X:{\mathcal B}(\mathbb{R}) \to{\mathcal E}(\Omega, {\mathcal A})\) satisfying (i) \(X(\mathbb{R})= I_\Omega\), (ii) \(X(\bigcup^\infty_{i=1} B_i)= \sum^\infty_{i=1} X(B_i)\), \(B_i\cap B_j= \emptyset\) for \(i\neq j\), the sum converging pointwise, i.e., by a fuzzy-event valued measure. The probability distribution of a fuzzy random variable \(X\) is the measure \(B\mapsto\int X(B)d\mu\). Standard random variables \(\widehat X:\Omega \to\mathbb{R}\) can be reformulated as measures \(X:{\mathcal B}(\mathbb{R}) \to{\mathcal E}(\Omega, {\mathcal A})\) according to \(X(B):= I_{X^{-1}(B)}\). It is proved that, conversely, every fuzzy random variable whose values are indicator functions on \(\Omega\), corresponds to a standard random variable \(\widehat X\) this way. Thus, standard random variables can be considered as very particular fuzzy random variables [for this and further results in this context, cf. the above cited paper by S. Bugajski et al.]. The author furthermore investigates joint fuzzy random variables as well as conditional expectations and independence of fuzzy random variables. Several theorems are proved, in particular, generalizations of the Borel-Cantelli lemma, the weak and the strong law of large numbers, and the central limit theorem in the fuzzy context. Summarizing, this paper presents an important relatively new subject concerning the foundations of probability theory as well as several results due to the author.