  Continued fractions have been introduced in the field of $p$--adic numbers
$\mathbb{Q}_p$ by several authors. However, a standard definition is still
missing since all the proposed algorithms are not able to replicate all the
properties of continued fractions in $\mathbb{R}$. In particular, an analogue
of the Lagrange's Theorem is not yet proved for any attempt of generalizing
continued fractions in $\mathbb{Q}_p$. Thus, it is worth to study the
definition of new algorithms for $p$--adic continued fractions. The main
condition that a new method needs to fulfill is the convergence in $\mathbb
Q_p$ of the continued fractions. In this paper we study some convergence
conditions for continued fractions in $\mathbb{Q}_p$. These results allow to
define many new families of continued fractions whose convergence is
guaranteed. Then we provide some new algorithms exploiting the new convergence
condition and we prove that one of them terminates in a finite number of steps
when the input is rational, as it happens for real continued fractions.
