What are examples of groups, where DLP (discrete logarithm problem) is hard?
Two obvious ones are: integers modulo $p$ ($p$ being prime) and elliptic curves over finite fields. What are the others?
As you probably already know, you can take discrete logarithms for any cyclic group. However as you can see on the linked wiki page, no algorithm is known for computing general discrete logarithms.
However some other popular choices of groups that are used for discrete logarithm are the algebraic torus over finite fields and the divisor class group of a curve over a finite field.