PyO3 supports two ways to define a free function in Python. Both require registering the function to a module.

One way is defining the function in the module definition.

use pyo3::prelude::*;

#[pymodule]
fn rust2py(py: Python, m: &PyModule) -> PyResult<()> {

    // Note that the `#[pyfn()]` annotation automatically converts the arguments from
    // Python objects to Rust values; and the Rust return value back into a Python object.
    #[pyfn(m, "sum_as_string")]
    fn sum_as_string_py(_py: Python, a:i64, b:i64) -> PyResult<String> {
       Ok(format!("{}", a + b))
    }

    Ok(())
}

fn main() {}

The other is annotating a function with #[pyfunction] and then adding it to the module using the wrap_pyfunction! macro.

use pyo3::prelude::*;
use pyo3::wrap_pyfunction;

#[pyfunction]
fn double(x: usize) -> usize {
    x * 2
}

#[pymodule]
fn module_with_functions(py: Python, m: &PyModule) -> PyResult<()> {
    m.add_wrapped(wrap_pyfunction!(double)).unwrap();

    Ok(())
}

fn main() {}

In order to make the function signature available to Python to be retrieved via inspect.signature, simply make sure the first line of your docstring is formatted like in the example below. Please note that the newline after the -- is mandatory. The / signifies the end of positional-only arguments. This is not a feature of this library in particular, but the general format used by CPython for annotating signatures of built-in functions. Function signatures for built-ins are new in Python 3 — in Python 2, they are simply considered to be a part of the docstring.

#![allow(unused_variables)]
fn main() {
use pyo3::prelude::*;

/// add(a, b, /)
/// --
///
/// This function adds two unsigned 64-bit integers.
#[pyfunction]
fn add(a: u64, b: u64) -> u64 {
    a + b
}
}

When annotated like this, signatures are also correctly displayed in IPython.

>>> pyo3_test.add?
Signature: pyo3_test.add(a, b, /)
Docstring: This function adds two unsigned 64-bit integers.
Type:      builtin_function_or_method

Currently, there are no conversions between Fns in Rust and callables in Python. This would definitely be possible and very useful, so contributions are welcome. In the meantime, you can do the following:

You can use ObjectProtocol::is_callable to check if you got a callable, which is true for functions (including lambdas), methods and objects with a __call__ method. You can call the object with ObjectProtocol::call with the args as first parameter and the kwargs (or None) as second parameter. There are also ObjectProtocol::call0 with no args and ObjectProtocol::call1 with only the positional args.

If you have a static function, you can expose it with #[pyfunction] and use wrap_pyfunction! to get the corresponding PyObject. For dynamic functions, e.g. lambda and functions that were passed as arguments, you must put them in some kind of owned container, e.g. a box. (A long-term solution will be a special container similar to wasm-bindgen's Closure). You can then use a #[pyclass] struct with that container as a field as a way to pass the function over the FFI barrier. You can even make that class callable with __call__ so it looks like a function in Python code.