We will reveal a natural correspondence between families of arcs originally studied by Nash and families of jets through the singular locus. Then we focus on normal locally complete intersection varieties with rational singularities. In particular, we introduce the notion of higher Du Val singularities, a higher dimensional version of Du Val singularities that preserve many of their features, and more generally, the notion of higher compound Du Val singularities, whose definition parallels that of compound Du Val singularities. For such singularities, we prove a one-to-one correspondence exists between families of arcs and families of jets through the singularities. As an application, we give a solution to the Nash problem for higher Du Val singularities.