There are two candidates for the role of the fundamental objects of M theory: a membrane and a five-brane. They are electric-magnetic dual to each other.

MEMBRANE:

With Vladimir Dotsenko and Christian Saemann we explore superconformal M2-brane actions based on three-algebras. We completely classify all relevant three-algebra representations in

On Superspace Actions for Multiple M2-Branes, Metric 3-Algebras and their Classification, Phys.Rev.D79:086002,2009,

and write the corresponding membrane actions. The first of these actions for a particular representation appears in

Multiple M2-branes and Generalized 3-Lie algebras, Phys.Rev.D78:066019,2008.

With Aki Hashimoto we find completely localized brane intersection solutions in

Supergravity solution of intersecting branes and AdS/CFT with flavor, JHEP 0211:036,2002,

identify the corresponding quantum gauge theories and use AdS/CFT correspondence to explore their dynamics.

FIVEBRANE:

I found supergravity solutions for partially localized five-brane intersections in

Supergravity solution for M5-brane intersection, arXiv:hep-th/9906203.

With John Schwarz we reformulate the Heterotic string action and demonstrate its equivalence to that of M theory five-brane wrapped on K3:

Wrapping the M theory five-brane on K3, Phys.Lett.B403:225-232,1997.