  Karo\'nski, {\L}uczak and Thomason conjectured in 2004 that for every finite
graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$
in such a way that the endvertices of each edge have different sums of incident
edge weights. This is known as the 1-2-3 Conjecture, and it was only recently
proved by Keusch. We extend this result to infinite graphs in the following
way. If $G$ is a graph without isolated edge, then the edges can be assigned
weights from $\{1,2,3\}$ is such a way that the endvertices of each edge have
different sum of incident edge weights, or these endvertices have both the same
infinite degree. We also investigate the extensions of theorems about total and
list versions of 1-2-3 Conjecture to infinite graphs.
