Laser-cooled trapped ions are among the most versatile experimental platforms for the simulation of non-trivial quantum Hamiltonians. What distinguishes this platform from others is the extent to which it is experimentally possible to control this system at the level of individual particles and interactions between them. Internal states of these ions, such as hyperfine states, constitute well isolated qubits (or spin-1/2 states) with long quantum coherence, and can be initialized as well as detected using laser beams with near perfection. Quantum logic gates and interacting spin Hamiltonians are engineered by coupling the spin states of multiple ions to their collective vibrational (phonon) modes using optical forces. By suitably tailoring these spin-phonon couplings, interactions between ion-spins can be tuned in magnitude, range, and sign, and in principle can be made arbitrary. In this talk, I will describe the quantum simulator that we are building at IQC, University of Waterloo to explore problems in quantum information and many-body physics in a regime that is intractable with classical computers.