We investigate the effective worldvolume theories of branes in a background given by (the bosonic sector of) 10-dimensional massive IIA supergravity (``massive branes'') and their M-theoretic origin. In the case of the solitonic 5-brane of type IIA superstring theory the construction of the Wess-Zumino term in the worldvolume action requires a dualization of the massive Neveu-Schwarz/Neveu-Schwarz target space 2-form field. We find that, in general, the effective worldvolume theory of massive branes contains new worldvolume fields that are absent in the massless case, i.e. when the mass parameter m of massive IIA supergravity is set to zero. We show how these new worldvolume fields can be introduced in a systematic way. In particular, we find new couplings between the massive solitonic 5-brane and the target space background, involving an additional worldvolume 1-form and 6-form. These new couplings have implications for the anomalous creation of branes. In particular, when a massive solitonic 5-brane passes through a D8-brane a stretched D6-brane is created. Similarly, in M-theory we find that when an M5-brane passes through an M9-brane a stretched Kaluza-Klein monopole is created. Pairs of massive branes of type IIA string theory can be viewed as the direct and double dimensional reduction of a single ``massive M-brane'' whose worldvolume theory is described by a gauged sigma model.

Latex file, 78 pages, 1 eps figure. Typos corrected. Version to be published in Nuclear Physics B