Summary: Given a general source \(X=\{X^n\}^\infty_{n=1}\), source coding is characterized by a pair \((\varphi_n,\psi_n)\) of encoder \(\varphi_n\) and decoder \(\psi_n\) together with the probability of error \[ \varepsilon_n \equiv \text{Pr}\biggl\{ \psi_n\bigl( \varphi_n(X^n) \bigr)\neq X^n\biggr\}. \] If the length of the encoder output \(\varphi_n (X^n)\) is fixed, then it is called fixed-length source coding, while if the length of the encoder output \(\varphi_n (X^n)\) is variable, then it is called variable-length source coding. Usually, in the context of fixed-length source coding the probability of error \(\varepsilon_n\) is required to asymptotically vanish (i.e., \(\lim_{n \to\infty} \varepsilon_n= 0)\), whereas in the context of variable-length source coding the probability of error \(\varepsilon_n\) is required to be exactly zero (i.e., \(\varepsilon_n =0\) \(\forall\;n=1,2, \dots)\). In contrast to these we consider in the present paper the problem of variable-length source coding with asymptotically vanishing probability of error (i.e., \(\lim_{n\to \infty} \varepsilon_n=0)\) and establish several fundamental theorems of this new subject.