  We are dealing with the question whether every group or semigroup action
(with some additional property) on a continuum (with some additional property)
has a fixed point. One of such results was given in 2009 by Shi and Sun. They
proved that every nilpotent group action on a uniquely arcwise connected
continuum has a fixed point. We are seeking for this type of results with e.g.
commutative, compact or torsion groups and semigroups acting on dendrites,
dendroids, $\lambda$-dendroids and uniquely arcwise connected continua. We
prove that every continuous action of a compact or torsion group on a uniquely
arcwise connected continuum has a fixed point. We also prove that every
continuous action of a compact and commutative semigroup on a uniquely arcwise
connected continuum or on a tree-like continuum has a fixed point.
