Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass $m$ into a finite or countably infinite number of masses with sum $m$, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes $x$ and $y$ runs the risk of a binary collision to form a single mass of magnitude $x+y$ at rate $\kappa(x,y)$, for some nonnegative, symmetric collision rate kernel $\kappa(x,y)$. Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable metric on the state space, and under appropriate restrictions on $\kappa$ and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Feller-like processes. A number of further results are obtained for the {\em additive coalescent} with collision kernel $\kappa(x,y) = x + y$. This process, which arises from the evolution of tree components in a random graph process, has asymptotic properties related to the stable subordinator of index $1/2$.