Some general remarks on improperly posed problems for partial differential equations -- Logarithmic convexity and other techniques applied to problems in continuum mechanics -- Cauchy's problem and the analytic continuation of solutions to elliptic equations -- Some properties of solutions of the Navier-Stokes equations -- Non-unique continuation for certain Ode's in Hilbert space and for uniformly parabolic and elliptic equations in self-adjoint divergence form -- Logarithmic convexity and the Cauchy problem for P(t)utt+M(t)ut+N(t)u=C in Hilbert space -- Stabilized quasi-reversibilite and other nearly-best-possible methods for non-well-posed problems.