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Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on the complex- and real-analytic side:

• A. Lewis - Notes on Global Analysis (link: https://mast.queensu.ca/~andrew/teaching/math942/)

According to the author these notes are no longer maintained, but the page had been linked to in MSE, which is why he hasn't removed it yet.

Having said that, concretely for complex- and real-analytic geometry there are more comprehensive sources. If you are interested in such, let me know and I will add them to the list.

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html) (Even though this topic is perhaps slightly more specific, I suggest it because there you can see more interactions with sheaf theory, and arguably distributions are among the central objects in Differential Geometry.)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005) (Chapter 4 is dedicated to the comparison of the various cohomologies, i.e Cech cohomology, Sheaf cohomology, Singular cohomology, de Rham cohomology. However, sheaves pop up in many of the other chapters as well, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. The last chapter includes elliptic complexes, some basic Hodge Theory, Kodaira's Vanishing Theorem etc. Of course, the latter topics are more comprehensively treated in books on complex geometry.)

A comprehensive introductory treatment specifically of sheaves on manifolds:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (As far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples. This would be my go-to recommendation for anyone who wants to learn about the sheaf-theoretic perspective of differentiable manifolds.)

Finally, there are the two volumes of

• Mallios - Geometry of Vector Sheaves, An Axiomatic Approach to Differential Geometry (1998),

which might be an interesting second read after getting acquinted with the basics.

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html) (Even though this topic is perhaps slightly more specific, I suggest it because there you can see more interactions with sheaf theory, and arguably distributions are among the central objects in Differential Geometry.)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005) (Chapter 4 is dedicated to the comparison of the various cohomologies, i.e Cech cohomology, Sheaf cohomology, Singular cohomology, de Rham cohomology. However, sheaves pop up in many of the other chapters as well, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. The last chapter includes elliptic complexes, some basic Hodge Theory, Kodaira's Vanishing Theorem etc. Of course, the latter topics are more comprehensively treated in books on complex geometry.)

A comprehensive introductory treatment specifically of sheaves on manifolds:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (As far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples. This would be my go-to recommendation for anyone who wants to learn about the sheaf-theoretic perspective of differentiable manifolds.)

Finally, there are the two volumes of

• Mallios - Geometry of Vector Sheaves, An Axiomatic Approach to Differential Geometry (1998),

which might be an interesting second read after getting acquinted with the basics.

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on the complex- and real-analytic side:

• A. Lewis - Notes on Global Analysis (link: https://mast.queensu.ca/~andrew/teaching/math942/, according to the author the notes are no longer maintained, but the page was linked to in MSE, which is why he hasn't removed the link).

Having said that, concretely for complex- and real-analytic geometry there are more comprehensive sources.

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005) (Sheaves pop up in many of the chapters, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. Chapter 4 is dedicated to the comparison of the various cohomologies etc.)

A comprehensive introductory treatment of sheaves on manifolds specifically:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (as far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples.)

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on the complex- and real-analytic side:

• A. Lewis - Notes on Global Analysis (link: https://mast.queensu.ca/~andrew/teaching/math942/)

According to the author these notes are no longer maintained, but the page had been linked to in MSE, which is why he hasn't removed it yet.

Having said that, concretely for complex- and real-analytic geometry there are more comprehensive sources. If you are interested in such, let me know and I will add them to the list.

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html) (Even though this topic is perhaps slightly more specific, I suggest it because there you can see more interactions with sheaf theory, and arguably distributions are among the central objects in Differential Geometry.)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005) (Chapter 4 is dedicated to the comparison of the various cohomologies, i.e Cech cohomology, Sheaf cohomology, Singular cohomology, de Rham cohomology. However, sheaves pop up in many of the other chapters as well, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. The last chapter includes elliptic complexes, some basic Hodge Theory, Kodaira's Vanishing Theorem etc. Of course, the latter topics are more comprehensively treated in books on complex geometry.)

A comprehensive introductory treatment specifically of sheaves on manifolds:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (As far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples. This would be my go-to recommendation for anyone who wants to learn about the sheaf-theoretic perspective of differentiable manifolds.)

Finally, there are the two volumes of

• Mallios - Geometry of Vector Sheaves, An Axiomatic Approach to Differential Geometry (1998),

which might be an interesting second read after getting acquinted with the basics.

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books:

With focus on the complex- and real-analytic side:

• A. Lewis - Notes on Global Analysis (link: https://mast.queensu.ca/~andrew/teaching/math942/, according to the author the notes are no longer maintained)

Having said that, concretely for complex- and real-analytic geometry there are more comprehensive sources.

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005)

A comprehensive introductory treatment of sheaves on manifolds specifically:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016)

Additionally to Bredon's and Warner's excellent books and Nicolaescu's lecture notes mentioned in ARA's answer and specifically for applications of sheaves in Differential Geometry I would suggest the following notes and books, in no particular order:

With focus on the complex- and real-analytic side:

• A. Lewis - Notes on Global Analysis (link: https://mast.queensu.ca/~andrew/teaching/math942/, according to the author the notes are no longer maintained, but the page was linked to in MSE, which is why he hasn't removed the link).

Having said that, concretely for complex- and real-analytic geometry there are more comprehensive sources.

With focus on distributions (in the sense of Differential Geometry):

• A. Lewis - Generalized Subbundles and Distributions, Comprehensive Review (2014) (link: https://mast.queensu.ca/~andrew/notes/abstracts/2011a.html)

With focus on analysis on manifolds:

• S. Ramanan - Global Calculus (2005) (Sheaves pop up in many of the chapters, including the last one on the consequences of the analysis of elliptic operators done in the previous chapter. Chapter 4 is dedicated to the comparison of the various cohomologies etc.)

A comprehensive introductory treatment of sheaves on manifolds specifically:

• T. Wedhorn - Manifolds, Sheaves, and Cohomology (2016) (as far as sheaf theory in differential geometry is concerned, this book is much more comprehensive than Warner's and contains nice examples.)