I will begin with a brief and biased historical survey of C*-algebras generated by isometries, discussing classical theorems of Coburn, Douglas, and Cuntz from the 60's and 70's. Although implicit from the beginning, the role of semigroup crossed products was not formalized until the late 80's and early 90's when Murphy, Stacey, Nica, and Raeburn and I used a covariance condition that works quite well for semigroups with LCMs. This was vastly generalized by Li in the 10's using constructible right ideals to define a universal semigroup algebra $C^*_s(P)$. I will then focus on my joint work with Sehnem of the 20's on a universal Toeplitz algebra $\mathcal T_u(P)$ defined via generators and relations when $P$ is a submonoid of a group $G$. When the semigroup satisfies an independence condition, $\mathcal T_u(P)$ coincides with $C^*_s(P)$, but it also behaves as expected when independence fails. For example, $\mathcal T_u(P)$ is the C*-algebra of the left regular representation for amenable $G$ and also in many nonamenable cases. We give characterizations of faithful representations and uniqueness, which are new even for right LCM monoids. I will also explain how Sehnem’s covariance algebra of a product system leads to a full boundary quotient of $\mathcal T_u(P)$, generalizing the boundary relations of quasi-lattice orders introduced by Crisp and myself in the 00's.