  We analyze and overview several different unconventional quantum
criticalities. One origin of the unconventionality is the proximity to
first-order transitions. The border between the first-order and continuous
transitions is described by a quantum tricritical point (QTCP) for
symmetry-breaking transitions. One of the characteristic features is the
concomitant divergence of order-parameter and uniform fluctuations in contrast
to the conventional quantum critical point (QCP). Several puzzling
non-Fermi-liquid properties are referred to be accounted for as in the cases of
YbRh2Si2, CeRu2Si2 and beta-YbAlB4. Another more dramatic unconventionality
appears in this case for topological transitions such as metal-insulator and
Lifshitz transitions. This border, the marginal quantum critical point (MQCP),
belongs to an unprecedented universality class with diverging uniform
fluctuations at zero temperature. The MQCP has a unique feature by a combined
character of symmetry-breaking and topological transitions. The theoretical
results are supported by experimental indications for V2-xCrxO3 and an organic
conductor kappa-(ET)2Cu[N(CN)2]Cl. Identifying topological transitions also
reveals how non-Fermi liquid appears as a phase in metals. The theory also
accounts for the criticality of a metamagnetic transition in ZrZn2, by
interpreting it as an interplay of Lifshitz transition and correlation effects.
We discuss common underlying physics in these examples.
