A useful feature of European and American options in the standard financial market model with constant coefficients is the property of put-call symmetry. This property states that the value of a put option with strike price K and maturity date T is the same as the value of a call option with strike price S, maturity date T in an auxiliary financial market with interest rate d and in which the underlying asset price pays dividends at the rate r and has initial value K. In this paper we review recent generalizations of this property and provide complementary results. We show taht put-call symmetry is a general property which holds in a large class of financial market models including nonmarkovian models with stochastic coefficients. The property extends naturally to nonstandard American claims such as (i) options with random maturity which include barrier options and capped options, (ii) multiasset derivatives, (iii) occupation time derivatives and (iv) claims whose payoffs are homogeneous of degree v is different from 1. Changes of numeraire which are instrumental in establishing symmetry properties are also reviewed and discussed.