Abstract: One essential feature of a scheme X which is flat and proper over a base scheme S is that for any other finite type S scheme, there is a finite type algebraic space Map(X,Y) parameterizing families of maps from X to Y. There have been several extensions of these results to the setting where X is a proper stack, and Y is a stack satisfying various hypotheses. Unfortunately many of the stacks arising in nature, such as global quotient stacks X/G, have affine stabilizer groups and are about as far as possible from being proper. However, we will show that for many non proper X and a large class of Y, the mapping stack Map(X,Y) is still algebraic and finite type. This leads us to introduce new notions of "projective" and "proper" for morphisms between stacks such that "projective" => "proper", and flat and "proper" => Map(X,Y) is algebraic for reasonable X. We discuss a large list of examples of "projective stacks", including X/G where G is reductive and X is projective-over-affine with H^0(O_X)^G finite dimensional, as well as any quotient stack which admits a projective good moduli space. Based on these, we will come up with an even longer list of "proper" stacks, including stacks which are proper over a scheme in the classical definition. Along the way, we will discuss some surprising "derived h-descent" results in derived algebraic geometry.