We study point and higher symmetries of systems of the
hydrodynamic type with and without an explicit dependence on
t,x. We consider such systems which satisfy the existence
conditions for an infinite-dimensional group of hydrodynamic
symmetries which implies linearizing transformations for these
systems. Under additional restrictions on the systems, we obtain
recursion operators for symmetries and use them to construct
infinite discrete sets of exact solutions of the studied
equations. We find the interrelation between higher symmetries and
recursion operators. Two-component systems are studied in more
detail than n-component systems. As a special case, we consider
Hamiltonian and semi-Hamiltonian systems of
Tsarëv.