  We develop a number of novel "black-bounce" spacetimes. These are specific
regular black holes where the "area radius" always remains non-zero, thereby
leading to a "throat" that is either timelike (corresponding to a traversable
wormhole), spacelike (corresponding to a "bounce" into a future universe), or
null (corresponding to a "one-way wormhole"). We shall first perform a general
analysis of the regularity conditions for such a spacetime, and then consider a
number of specific examples. The examples are constructed using a mass function
similar to that of Fan--Wang, and fall into several particular cases, such as
the original Simpson--Visser model, a Bardeen-type model, and other
generalizations thereof. We shall analyse the regularity, the energy
conditions, and the causal structure of these models. The main results are
several new geometries, more complex than before, with two or more horizons,
with the possibility of an extremal case. We shall derive a general theorem
regarding static space-time regularity, and another general theorem regarding
(non)-satisfaction of the classical energy conditions.
