Example bridging header generated by a simple Swift file. Tons of definition cruft up top; the heart of it is a few lines at the end.

CallMeFromObjC.h

// Generated by Apple Swift version 3.0.1 (swiftlang-800.0.58.6 clang-800.0.42.1)
#pragma clang diagnostic push

#if defined(__has_include) && __has_include(<swift/objc-prologue.h>)
# include <swift/objc-prologue.h>
#endif

#pragma clang diagnostic ignored "-Wauto-import"
#include <objc/NSObject.h>
#include <stdint.h>
#include <stddef.h>
#include <stdbool.h>

#if !defined(SWIFT_TYPEDEFS)
# define SWIFT_TYPEDEFS 1
# if defined(__has_include) && __has_include(<uchar.h>)
#  include <uchar.h>
# elif !defined(__cplusplus) || __cplusplus < 201103L
typedef uint_least16_t char16_t;
typedef uint_least32_t char32_t;
# endif
typedef float swift_float2  __attribute__((__ext_vector_type__(2)));
typedef float swift_float3  __attribute__((__ext_vector_type__(3)));
typedef float swift_float4  __attribute__((__ext_vector_type__(4)));
typedef double swift_double2  __attribute__((__ext_vector_type__(2)));
typedef double swift_double3  __attribute__((__ext_vector_type__(3)));
typedef double swift_double4  __attribute__((__ext_vector_type__(4)));
typedef int swift_int2  __attribute__((__ext_vector_type__(2)));
typedef int swift_int3  __attribute__((__ext_vector_type__(3)));
typedef int swift_int4  __attribute__((__ext_vector_type__(4)));
typedef unsigned int swift_uint2  __attribute__((__ext_vector_type__(2)));
typedef unsigned int swift_uint3  __attribute__((__ext_vector_type__(3)));
typedef unsigned int swift_uint4  __attribute__((__ext_vector_type__(4)));
#endif

#if !defined(SWIFT_PASTE)
# define SWIFT_PASTE_HELPER(x, y) x##y
# define SWIFT_PASTE(x, y) SWIFT_PASTE_HELPER(x, y)
#endif
#if !defined(SWIFT_METATYPE)
# define SWIFT_METATYPE(X) Class
#endif
#if !defined(SWIFT_CLASS_PROPERTY)
# if __has_feature(objc_class_property)
#  define SWIFT_CLASS_PROPERTY(...) __VA_ARGS__
# else
#  define SWIFT_CLASS_PROPERTY(...)
# endif
#endif

#if defined(__has_attribute) && __has_attribute(objc_runtime_name)
# define SWIFT_RUNTIME_NAME(X) __attribute__((objc_runtime_name(X)))
#else
# define SWIFT_RUNTIME_NAME(X)
#endif
#if defined(__has_attribute) && __has_attribute(swift_name)
# define SWIFT_COMPILE_NAME(X) __attribute__((swift_name(X)))
#else
# define SWIFT_COMPILE_NAME(X)
#endif
#if defined(__has_attribute) && __has_attribute(objc_method_family)
# define SWIFT_METHOD_FAMILY(X) __attribute__((objc_method_family(X)))
#else
# define SWIFT_METHOD_FAMILY(X)
#endif
#if defined(__has_attribute) && __has_attribute(noescape)
# define SWIFT_NOESCAPE __attribute__((noescape))
#else
# define SWIFT_NOESCAPE
#endif
#if !defined(SWIFT_CLASS_EXTRA)
# define SWIFT_CLASS_EXTRA
#endif
#if !defined(SWIFT_PROTOCOL_EXTRA)
# define SWIFT_PROTOCOL_EXTRA
#endif
#if !defined(SWIFT_ENUM_EXTRA)
# define SWIFT_ENUM_EXTRA
#endif
#if !defined(SWIFT_CLASS)
# if defined(__has_attribute) && __has_attribute(objc_subclassing_restricted)
#  define SWIFT_CLASS(SWIFT_NAME) SWIFT_RUNTIME_NAME(SWIFT_NAME) __attribute__((objc_subclassing_restricted)) SWIFT_CLASS_EXTRA
#  define SWIFT_CLASS_NAMED(SWIFT_NAME) __attribute__((objc_subclassing_restricted)) SWIFT_COMPILE_NAME(SWIFT_NAME) SWIFT_CLASS_EXTRA
# else
#  define SWIFT_CLASS(SWIFT_NAME) SWIFT_RUNTIME_NAME(SWIFT_NAME) SWIFT_CLASS_EXTRA
#  define SWIFT_CLASS_NAMED(SWIFT_NAME) SWIFT_COMPILE_NAME(SWIFT_NAME) SWIFT_CLASS_EXTRA
# endif
#endif

#if !defined(SWIFT_PROTOCOL)
# define SWIFT_PROTOCOL(SWIFT_NAME) SWIFT_RUNTIME_NAME(SWIFT_NAME) SWIFT_PROTOCOL_EXTRA
# define SWIFT_PROTOCOL_NAMED(SWIFT_NAME) SWIFT_COMPILE_NAME(SWIFT_NAME) SWIFT_PROTOCOL_EXTRA
#endif

#if !defined(SWIFT_EXTENSION)
# define SWIFT_EXTENSION(M) SWIFT_PASTE(M##_Swift_, __LINE__)
#endif

#if !defined(OBJC_DESIGNATED_INITIALIZER)
# if defined(__has_attribute) && __has_attribute(objc_designated_initializer)
#  define OBJC_DESIGNATED_INITIALIZER __attribute__((objc_designated_initializer))
# else
#  define OBJC_DESIGNATED_INITIALIZER
# endif
#endif
#if !defined(SWIFT_ENUM)
# define SWIFT_ENUM(_type, _name) enum _name : _type _name; enum SWIFT_ENUM_EXTRA _name : _type
# if defined(__has_feature) && __has_feature(generalized_swift_name)
#  define SWIFT_ENUM_NAMED(_type, _name, SWIFT_NAME) enum _name : _type _name SWIFT_COMPILE_NAME(SWIFT_NAME); enum SWIFT_COMPILE_NAME(SWIFT_NAME) SWIFT_ENUM_EXTRA _name : _type
# else
#  define SWIFT_ENUM_NAMED(_type, _name, SWIFT_NAME) SWIFT_ENUM(_type, _name)
# endif
#endif
#if !defined(SWIFT_UNAVAILABLE)
# define SWIFT_UNAVAILABLE __attribute__((unavailable))
#endif
#if defined(__has_feature) && __has_feature(modules)
@import ObjectiveC;
#endif

#pragma clang diagnostic ignored "-Wproperty-attribute-mismatch"
#pragma clang diagnostic ignored "-Wduplicate-method-arg"

SWIFT_CLASS("_TtC9Bridgette14CallMeFromObjC")
@interface CallMeFromObjC : NSObject
@property (nonatomic, copy) NSString * _Nonnull name;
- (nonnull instancetype)initWithName:(NSString * _Nonnull)name OBJC_DESIGNATED_INITIALIZER;
- (void)speak;
- (nonnull instancetype)init SWIFT_UNAVAILABLE;
@end

#pragma clang diagnostic pop

CallMeFromObjC.swift

// You have to import Foundation to be able to @objc anything.
import Foundation

// And a class has to inherit from NSObject to be visible to Obj-C.
// (If you wanna write a root class, you'll have to do it in Obj-C.)
public class CallMeFromObjC: NSObject {
    // Publish the API you want to use from Obj-C.
    public var name: String

    public init(name: String) {
        self.name = name
    }

    public func speak() {
        print("\(self)'s name is: \(name)")
    }
}


// To compile, run:
//
//     install -d build
//     xcrun -sdk macosx10.12 swift -frontend -c -primary-file CallMeFromObjC.swift -sdk /Applications/Xcode.app/Contents/Developer/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.12.sdk/ -module-name Bridgette -emit-module-path build/Bridgette.swiftmodule -emit-objc-header-path build/CallMeFromObjC.h -enable-testing -enable-objc-interop -parse-as-library -o build/CallMeFromObjC.o