Like their classical counterparts, tropical varieties come together
with their homology groups. These homology groups are bigraded
(one can talk of (p,q)-cycles) and correspond to degenerations of
the homology groups of families of complex varieties approximating
tropical varieties if such approximations exist (but are defined even
in the absence of such approximating families). E.g. algebraic
cycles degenerate to a particularly nice type of tropical (p,p)-cycles
(though very little is known of algebraically realizable tropical cycles,
unless p=1 or p=n-1). Furthermore, a similar-looking type of tropical
cycles admit a somewhat different type of liftings, such as monodromies
in homology. The talk will review some known results as well as some
work in progress -- in particular joint work in progress with Brugalle
and with Itenberg, Katzarkov and Zharkov.