  Combining the definition of 0-Hecke monoids with that of Kiselman semigroups,
we define what we call Kiselman quotients of 0-Hecke monoids associated with
simply laced Dynkin diagrams. We classify these monoids up to isomorphism,
determine their idempotents and show that they are $\mathcal{J}$-trivial. For
type $A$ we show that Catalan numbers appear as the maximal cardinality of our
monoids, in which case the corresponding monoid is isomorphic to the monoid of
all order-preserving and order-decreasing total transformations on a finite
chain. We construct various representations of these monoids by matrices, total
transformations and binary relations. Motivated by these results, with a mixed
graph we associate a monoid, which we call a Hecke-Kiselman monoid, and
classify such monoids up to isomorphism. Both Kiselman semigroups and Kiselman
quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.
