Re: cross breeding plants from different genus groups

It is certainly possible to cross breed plants from different genera but it 
is not something that occurs with great frequency even with human 
intervention.  Cross breeding different genera has been done.  For example, 
crosses between radish (Raphanus sativus) and cabbage (Brassica oleracea) 
have been made.  There has been some success in ultimately getting some 
fertile offspring.  Unfortunately the offspring have been useless since the 
leaves are like a radish and the roots are like a cabbage (ideally the roots 
would be like radish and the leaves like cabbage).  So, yes, it is possible 
to do this, but this is more of an exception than a rule.  It depends on the 
plants you are trying to cross breed.  Some combinations would be more 
likely than others.

In terms of the complexity of the process, the cross breeding itself is not 
difficult and can be done by the home gardener.  It is a matter of making 
sure that both plants are flowering.  Pollen from the anther of one flower 
would need to be transferred to the stigma of the other flower.  The anther 
is often though not always yellow.  The stigma is in the very center of the 
flower and represents part of the female reproductive structure in flowering 
plants.

It is best to remove the anthers of the flowers carefully...you have to make 
sure you do not get any pollen of the flower onto its own stigma.  If you do 
that you will not be doing any cross breeding.  Typical gardening books have 
pictures of flowers and how to make hybrids so you might want to check out 
those types of books.

Sources:
Klug and Cummings.  2000.  Concepts of Genetics, 6th edition.  Prentice Hall


David Hershey adds:

The family that most easily lends itself to intergeneric hybrids is the Orchid 
family. Intergeneric orchid hybrids are widely grown. There are dozens of 
examples of bigeneric orchids, such as x Laeliocattleya  
(Cattleya  x Laelia) and x Brassocattleya  
(Brassavola  x Cattleya )  . 

The x in front of the genus name is pronounced "hybrid genus."

If you want to grow a bigeneric plant, try the common houseplant Botanical 
Wonder (x Fatshedera lizei) or tree, Leyland Cypress (x 
Cuppressocyparis leylandii).