The classical output regulation problem formulation for linear systems has a number of shortcomings; among them a primary one is that it does not take into account the transient response. Although this problem has been studied since the 1970s, a complete picture has not emerged yet. We formulate and study here a number of output regulation problems which incorporate the notion of transient performance into them. By defining a performance index that involves transient response, we define optimal and suboptimal control problems which are constrained by the steady state output regulation requirement. Such constrained optimal and suboptimal control problems for a given system are studied by transforming them to unconstrained optimal and suboptimal control problems for certain auxiliary systems. Several issues such as obtaining expressions for the infimal performance measure, solvability conditions, regulator construction are studied in detail. Both state and measurement feedback controllers are considered.