Microscope for super-resolution scanning microscopy of a specimen, comprising - an illumination device for illuminating the specimen, comprising an illumination beam path which guides illumination light via a principal colour splitter, a scanner and an objective to the specimen, - an imaging device for imaging a point spot or line spot on the specimen into a diffraction-limited individual image at rest under an imaging scale into a detection plane, wherein the imaging device has an imaging beam path which images specimen light via the objective, the scanner and the principal colour splitter into the detection plane, - a detection device for capturing the individual image in the detection plane for different scanning positions of the scanner with a spatial resolution which, taking account of the imaging scale in at least one extent/dimension, is at least twice as large as a full width at half maximum of the diffraction-limited individual image, - an evaluation device for evaluating a diffraction structure of the individual image for the scanning positions from the data of the detection device, and for generating an image of the specimen with a resolution increased beyond the diffraction limit, characterized in that - a phase mask (48.1, 49.1) with a laterally varying profile of influencing the phase is arranged in the illumination beam path, is passed through by all of the illumination light, is arranged in or in the vicinity of the objective pupil or a plane conjugate thereto and is designed to generate an illumination pattern in the specimen that differs from the point spot or line spot, the different illumination pattern overlapping with the point spot or line spot such that the structure of the illumination pattern is located within the region captured by the detection device, wherein - either the phase mask (48.1) only acts on the illumination light or - the phase mask (49.1) acts on the illumination light and the specimen light, and a mask (48.1) with a phase structure inverted to that of the phase mask (49.1) is provided and only acts on the specimen light and cancels the effect of the phase mask (49.1), wherein the illumination pattern is formed by a plurality of spatially separated illumination spots (S1, S2) lying next to one another or by an illumination light distribution in the form of a donut ring. - an illumination device for illuminating the specimen, comprising an illumination beam path which guides illumination light via a principal colour splitter, a scanner and an objective to the specimen, - an imaging device for imaging a point spot or line spot on the specimen into a diffraction-limited individual image at rest under an imaging scale into a detection plane, wherein the imaging device has an imaging beam path which images specimen light via the objective, the scanner and the principal colour splitter into the detection plane, - a detection device for capturing the individual image in the detection plane for different scanning positions of the scanner with a spatial resolution which, taking account of the imaging scale in at least one extent/dimension, is at least twice as large as a full width at half maximum of the diffraction-limited individual image, - an evaluation device for evaluating a diffraction structure of the individual image for the scanning positions from the data of the detection device, and for generating an image of the specimen with a resolution increased beyond the diffraction limit, characterized in that - a phase mask (48.1, 49.1) with a laterally varying profile of influencing the phase is arranged in the illumination beam path, is passed through by all of the illumination light, is arranged in or in the vicinity of the objective pupil or a plane conjugate thereto and is designed to generate an illumination pattern in the specimen that differs from the point spot or line spot, the different illumination pattern overlapping with the point spot or line spot such that the structure of the illumination pattern is located within the region captured by the detection device, wherein - either the phase mask (48.1) only acts on the illumination light or - the phase mask (49.1) acts on the illumination light and the specimen light, and a mask (48.1) with a phase structure inverted to that of the phase mask (49.1) is provided and only acts on the specimen light and cancels the effect of the phase mask (49.1), wherein the illumination pattern is formed by a plurality of spatially separated illumination spots (S1, S2) lying next to one another or by an illumination light distribution in the form of a donut ring. - either the phase mask (48.1) only acts on the illumination light or - the phase mask (49.1) acts on the illumination light and the specimen light, and a mask (48.1) with a phase structure inverted to that of the phase mask (49.1) is provided and only acts on the specimen light and cancels the effect of the phase mask (49.1),