  Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension
of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That
is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to
$L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose
of this paper is to present a refined version of MacLane's theory of key
polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of
related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo.
  Namely, we associate to $\iota$ a countable well ordered set $$
\mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; $$ the $Q_i$ are called {\bf key
polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are
called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$.
  We give an explicit description of the limit key polynomials (which may be
viewed as a generalization of the Artin--Schreier polynomials). We also give an
upper bound on the order type of the set of key polynomials. Namely, we show
that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has
order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$
this order type is bounded above by $\omega\times\omega$, where $\omega$ stands
for the first infinite ordinal.
