  I explore two separate topics: the concept of jointness for set-theoretic
guessing principles, and the notion of grounded forcing axioms. A family of
guessing sequences is said to be joint if all of its members can guess any
given family of targets independently and simultaneously. I primarily
investigate jointness in the case of various kinds of Laver diamonds. In the
case of measurable cardinals I show that, while the assertions that there are
joint families of Laver diamonds of a given length get strictly stronger with
increasing length, they are all equiconsistent. This is contrasted with the
case of partially strong cardinals, where we can derive additional consistency
strength, and ordinary diamond sequences, where large joint families exist
whenever even one diamond sequence does. Grounded forcing axioms modify the
usual forcing axioms by restricting the posets considered to a suitable ground
model. I focus on the grounded Martin's axiom which states that Martin's axioms
holds for posets coming from some ccc ground model. I examine the new axiom's
effects on the cardinal characteristics of the continuum and show that it is
quite a bit more robust under mild forcing than Martin's axiom itself.
