ESP 8266 does not respond at all
Start by uploading blink sketch to your Arduino then connect it to your ESP8266 like this: TX-TX and RX-RX.

Now open Serial Monitor and send AT command and see if it responds. If so then you can control it using your Arduino by wiring it back to TX-RX and RX-TX.

PS: Don't forget to set the line ending in the Serial Monitor to Newline or Carriage Return.

Edit: Use this code to connect to ESP8266:

#include &lt;SoftwareSerial.h&gt;

const byte rxPin = 2; // Wire this to Tx Pin of ESP8266
const byte txPin = 3; // Wire this to Rx Pin of ESP8266

// We'll use a software serial interface to connect to ESP8266
SoftwareSerial ESP8266 (rxPin, txPin);

void setup() {
  Serial.begin(115200);
  ESP8266.begin(115200); // Change this to the baudrate used by ESP8266
  delay(1000); // Let the module self-initialize
}

void loop() {
  Serial.println("Sending an AT command...");
  ESP8266.println("AT");
  delay(30);
  while (ESP8266.available()){
     String inData = ESP8266.readStringUntil('\n');
     Serial.println("Got reponse from ESP8266: " + inData);
  }  
}


Since the Hardware Serial interface will be busy when connected to Computer, then you have to use another Serial interface to communicate with your ESP8266. In this case, Software Serial comes handy.