We define being Lindel?f with respect to an ideal and investigate basic properties of the concept, its relation to known concepts, and its preservation by functions, suaces, pre-images, and products. An ideal is a nonempty collection of subsets of X closed under operations of subset ( “heredity” ) and finite union ( “finite additivity” ). If in addition the ideal is closed under the operation of countable unions, it is called ? - ideal. We denote a topological space (X,?) with an ideal I defined on X as (X,?,I) and call (X,?,I) an ideal topological space. A space (X,?,I) is said to be I - Lindel?f or Lindel?f with respect to I, if every open cover U of X has a countable subcollection V such that X ? ?V ? I. a space is Lindel?f iff it is {?} - Lindel?f. Frolik defines a space to be weakly Lindel?f if every open cover U of the space has a countable subcollection V such that X = ?V. We now show that weakly Lindel?f spaces are a special case of Lindel?f with respect to an ideal. If (X,?) is a space, we denote the ideal ofnowhere dense setsbyN(?) and the? -ideal. of meager( first category ) subsetsbyM(?). An ideal I on (X,?) is said to be ? - codense if I ? ? = ?. Let (X,?) be a space. (1) (X,?) is weakly Lindel?f iff (X,?) is N(?) - Lindel?f. (2) (X,?) is weakly Lindel?f iff (X,?) is Lindel?f with respect to some ? - codense ideal. (3) If (X,?) is a Baire space, then (X,?) is weakly Lindel?f iff (X,?) is M(?) - Lindel?f. A space X is said to be (countably) compact with respect to an ideal I, or simply (countably) I- compact, if every (countable) open cover of the space admits a finite subcollection which covers all the space except for a set in the ideal. It is shown that a space X with an ideal I is countably I - compact if and only if every locally finite collection of non-ideal subsets is finite. A space X with an ideal I is said to be paracompact with respect to I, or simply I-paracompact, if every open cover of the space admits a locally finite open refinement (not necessarily a cover) which covers all the space except for a set in the ideal. Let (X,?) be a space with an ideal I such that X is I-paracompact and I ? ? = {?}. It is then shown that countable I-compactness and I-compactness are equivalent. Special cases include; countable compactness is equivalent to compactness in paracompact spaces; light compactness is equivalent to quasiH-closedness in almost paracompact spaces; and countable meager- compactness is equivalent to meager-compactness in meager-paracompact Baire spaces.