  We introduce a new approach to create and detect Majorana fermions using
optically trapped 1D fermionic atoms. In our proposed setup, two internal
states of the atoms couple via an optical Raman transition---simultaneously
inducing an effective spin-orbit interaction and magnetic field---while a
background molecular BEC cloud generates s-wave pairing for the atoms. The
resulting cold atom quantum wire supports Majorana fermions at phase boundaries
between topologically trivial and nontrivial regions, as well as `Floquet
Majorana fermions' when the system is periodically driven. We analyze
experimental parameters, detection schemes, and various imperfections.
