  In this paper, we introduce {\it factorial} analogues of the ordinary
Hall--Littlewood $P$- and $Q$-polynomials, which we call the {\it factorial
Hall--Littlewood $P$- and $Q$-polynomials}. Using the {\it universal} formal
group law, we further generalize these polynomials to the {\it universal
factorial Hall--Littlewood $P$- and $Q$-functions}.
  We show that these functions satisfy the {\it vanishing property} which the
ordinary factorial Schur $S$-, $P$-, and $Q$-polynomials have. By the vanishing
property, we derive the Pieri-type formula and a certain generalization of the
classical hook formula.
  We then characterize our functions in terms of Gysin maps from flag bundles
in the complex cobordism theory. Using this characterization and Gysin formulas
for flag bundles, we can obtain generating functions for the universal
factorial Hall--Littlewood $P$- and $Q$-functions. Using our generating
functions, we can show that our factorial Hall--Littlewood $P$- and
$Q$-polynomials have a certain {\it cancellation property}.
  Further applications such as Pfaffian formulas for $K$-theoretic factorial
$Q$-polynomials are also given.
