The Toeplitz C*-algebra $\mathcal{T}_\lambda(P)$ of a left cancellative monoid $P$ is the C*-algebra generated by the left regular representation of $P$ by isometries on $\ell^2(P)$. In this talk I will report on joint work with M. Laca on Toeplitz C*-algebras associated to submonoids of groups. We will focus on faithfulness of representations and discuss examples coming from orders in algebraic number fields. We provide a presentation for the full analogue of the boundary quotient $\partial\mathcal{T}_\lambda(P)$ in terms of a new notion of foundation sets, and give sufficient (sometimes also necessary) conditions on $P$ for the boundary quotient to be purely infinite simple. We will discuss applications of these results to C*-algebras associated to certain submonoids of the Thompson group F. This last part is based on joint work in progress with A. an Huef, M. Laca, B. Nucinkis, and I. Raeburn.