  The paper addresses questions of existence and regularity of solutions to
linear partial differential equations whose coefficients are generalized
functions or generalized constants in the sense of Colombeau. We introduce
various new notions of ellipticity and hypoellipticity, study their
interrelation, and give a number of new examples and counterexamples. Using the
concept of $\G^\infty$-regularity of generalized functions, we derive a general
global regularity result in the case of operators with constant generalized
coefficients, a more specialized result for second order operators, and a
microlocal regularity result for certain first order operators with variable
generalized coefficients. We also prove a global solvability result for
operators with constant generalized coefficients and compactly supported
Colombeau generalized functions as right hand sides.
