Sending json array via Alamofire

You can just encode the JSON with NSJSONSerialization and then build the NSURLRequest yourself. For example, in Swift 3:

var request = URLRequest(url: url)
request.httpMethod = "POST"
request.setValue("application/json", forHTTPHeaderField: "Content-Type")

let values = ["06786984572365", "06644857247565", "06649998782227"]

request.httpBody = try! JSONSerialization.data(withJSONObject: values)

AF.request(request)                               // Or `Alamofire.request(request)` in prior versions of Alamofire
    .responseJSON { response in
        switch response.result {
        case .failure(let error):
            print(error)
            
            if let data = response.data, let responseString = String(data: data, encoding: .utf8) {
                print(responseString)
            }
        case .success(let responseObject):
            print(responseObject)
        }
}

For Swift 2, see previous revision of this answer.

For swift 3 and Alamofire 4 I use the following ParametersEncoding and Array extension:

import Foundation
import Alamofire

private let arrayParametersKey = "arrayParametersKey"

/// Extenstion that allows an array be sent as a request parameters
extension Array {
    /// Convert the receiver array to a `Parameters` object. 
    func asParameters() -> Parameters {
        return [arrayParametersKey: self]
    }
}


/// Convert the parameters into a json array, and it is added as the request body. 
/// The array must be sent as parameters using its `asParameters` method.
public struct ArrayEncoding: ParameterEncoding {

    /// The options for writing the parameters as JSON data.
    public let options: JSONSerialization.WritingOptions


    /// Creates a new instance of the encoding using the given options
    ///
    /// - parameter options: The options used to encode the json. Default is `[]`
    ///
    /// - returns: The new instance
    public init(options: JSONSerialization.WritingOptions = []) {
        self.options = options
    }

    public func encode(_ urlRequest: URLRequestConvertible, with parameters: Parameters?) throws -> URLRequest {
        var urlRequest = try urlRequest.asURLRequest()

        guard let parameters = parameters,
            let array = parameters[arrayParametersKey] else {
                return urlRequest
        }

        do {
            let data = try JSONSerialization.data(withJSONObject: array, options: options)

            if urlRequest.value(forHTTPHeaderField: "Content-Type") == nil {
                urlRequest.setValue("application/json", forHTTPHeaderField: "Content-Type")
            }

            urlRequest.httpBody = data

        } catch {
            throw AFError.parameterEncodingFailed(reason: .jsonEncodingFailed(error: error))
        }

        return urlRequest
    }
}

Basically, it converts the array to a Dictionary in order to be accepted as Parameters argument, and then it takes back the array from the dictionary, convert it to JSON Data and adds it as the request body.

Once you have it, you can create request this way:

let values = ["06786984572365", "06644857247565", "06649998782227"]
Alamofire.request(url,
                  method: .post,
                  parameters: values.asParameters(),
                  encoding: ArrayEncoding())