We study the pricing and hedging of contingent claims in a Markov regime-switching market with a money market account, a zero-coupon bond, and an ordinary share. General contingent claims with payoffs depending on both the share price and the state of a Markov chain describing regime switching are considered. A general pricing kernel defined by the product of two density processes is used to explicitly take into account regime switching risk. Under some differentiability and boundedness conditions, a martingale representation result is established and the integrands in the representation are explicitly identified with respect to the general pricing kernel. We then determine a pricing kernel and a hedging strategy by minimizing the residual risk due to incomplete hedging. Our analysis is also extended to Asian-style and American-style general contingent claims.