Transseries' is the short name coined by Ecalle for certain generalised power series - here the name is appropriate to denote all such series. The use of transeries allows the study of o-minimal expansions of the reals to rest on fully model theoretic methods. We develop this theme in case the o-minimal expansion of the reals is polynomially bounded and we prove results of quantifier(s) elimination, of axiomatisation, of cell decomposition and relative computability - in the restricted case' where the primitive functions added to the real field are C^\infty with arguments ranging over [0,1].