Hamiltonian simulation (HS) is the problem of simulating the dynamics of a quantum system on a quantum computer. In numerous cases, interest lies in instances where the initial state is supported within a low-energy sector of the Hamiltonian. In this talk, I will discuss new quantum algorithms that use this low-energy condition to speedup generic HS approaches and identify the class of HS problems that can benefit from these algorithms. I will also show matching lower bounds in terms of query and gate complexities that demonstrate the optimality of these algorithms.