  We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$
is the minimum number of colours in a vertex colouring of $G$ that is preserved
only by the trivial automorphism. An analogous invariant for edge colourings is
called the distinguishing index, denoted by $D'(G)$. We prove that $D'(G)\leq
D(G)+1$. For proper colourings, we study relevant invariants called the
distinguishing chromatic number $\chi_D(G)$, and the distinguishing chromatic
index $\chi'_D(G)$, for vertex and edge colourings, respectively. We show that
$\chi_D(G)\leq 2\Delta(G)-1$ for graphs with a finite maximum degree
$\Delta(G)$, and we obtain substantially lower bounds for some classes of
graphs with infinite motion. We also show that $\chi'_D(G)\leq \chi'(G)+1$,
where $\chi'(G)$ is the chromatic index of $G$, and we prove a similar result
$\chi''_D(G)\leq \chi''(G)+1$ for proper total colourings. A number of
conjectures are formulated.
