Abstract: We will describe the connective K-cohomology of U(n), Sp(n), O(n), their oriented variants and the torus T(n) and the 'symplectic torus' Sp(1)^n, in so far as it is understood at present. Of particular oimportance is the relation to representation theory and cohomology. The symplectic splitting principle and the role of Pontrjagin classes are central. The best results are available for ku^* and the more highly connected groups (e.g., Sp(n) rather than O(n)), but some results are available for ko^* as well. We will start with some general facts about the relations between connective K-theory, representation theory, and cohomology.