An ascent sequence is one consisting of non-negative integers in which the size of each letter is restricted by the number of ascents preceding it in the sequence. Ascent sequences have recently been shown to be related to (2+2)-free posets and a variety of other combinatorial structures. Let Fn denote the Fibonacci sequence given by the recurrence $F_n=F_{n-1}+F_{n-2}$ if $n\geq2$, with $F_0=0$ and $F_1=1$. In this paper, we draw connections between ascent sequences and the Fibonacci numbers by showing that several pattern-avoidance classes of ascent sequences are enumerated by either $F_{n+1}$ or $F_{2n-1}$. We make use of both algebraic and combinatorial methods to establish our results. In one of the apparently more difficult cases, we make use of the \emph{kernel method} to solve a functional equation and thus determine the distribution of some statistics on the avoidance class in question. In two other cases, we adapt the \emph{scanning-elements algorithm}, a technique which has been used in the enumeration of certain classes of pattern-avoiding permutations, to the comparable problem concerning pattern-avoiding ascent sequences.