user48341: I managed to get the LowPower libary working on my Arduino Micro. This code works perfectly:
#include <LowPower.h>

void setup() {
  // initialize digital pin LED_BUILTIN as an output.
  pinMode(LED_BUILTIN, OUTPUT);
}

// the loop function runs over and over again forever
void loop() {
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}

However. I also tried it with a serial connection, to put out some text periodically. But I think the lowpower-libary does - in order to save power - disable this function. So now I am worried that it also disables some other things I might have to use but won't work. Like OneWire or I2C. It also would be nice to work with a functioning standby-mode while developing where serial output is vital, altough it's not neccessary, because I can simulate it with a simple delay. So my question is: How can I use the lowpower-libary AND have a serial connection. And will this libary affect OneWire or I2C? If so: Is there a way on how to keep these "services" running? Or is there maybe another libary wich also saves power in a similar way, but where you have more abilities to customize what you want to have shut down?

user4143: Yes, activating low power does disable the UART - but only while it's sleeping.
To use serial you just need to ensure that you only use it while it's not sleeping - and due to the interrupt and buffered nature of the Serial driver it's not as easy as just printing.
However, it's simple to achieve: just flush the serial before sleeping. This ensures that any pending serial data is sent out through the serial port before you go to sleep:
void loop() {
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}

If that doesn't help you may need to turn serial off and on yourself either side of your powerdown:
void loop() {
  Serial.begin(115200);
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.begin(115200);
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}


Update: 
I just noticed you're using the Micro. That makes a big difference. The Micro and the Leonardo's Serial object is actually a USB CDC/ACM serial emulation, not UART.  You cannot sustain a USB connection while sleeping. So it will be killed.
To use communication while sleeping you will have to switch to a proper UART connection with a USB to TTL UART (e.g., FT232RL) adaptor to connect to the TX/RX pins and use Serial1 instead.

48341: <p>I managed to get the LowPower libary working on my Arduino Micro. This code works perfectly:</p>

<pre><code>#include &lt;LowPower.h&gt;

void setup() {
  // initialize digital pin LED_BUILTIN as an output.
  pinMode(LED_BUILTIN, OUTPUT);
}

// the loop function runs over and over again forever
void loop() {
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}
</code></pre>

<p>However. I also tried it with a serial connection, to put out some text periodically. But I think the lowpower-libary does - in order to save power - disable this function. So now I am worried that it also disables some other things I might have to use but won't work. Like OneWire or I2C. It also would be nice to work with a functioning standby-mode while developing where serial output is vital, altough it's not neccessary, because I can simulate it with a simple delay. So my question is: How can I use the lowpower-libary AND have a serial connection. And will this libary affect OneWire or I2C? If so: Is there a way on how to keep these "services" running? Or is there maybe another libary wich also saves power in a similar way, but where you have more abilities to customize what you want to have shut down?</p>
 4143: <p>Yes, activating low power does disable the UART - but only <em>while</em> it's sleeping.</p>

<p>To use serial you just need to ensure that you only use it while it's not sleeping - and due to the interrupt and buffered nature of the Serial driver it's not as easy as just printing.</p>

<p>However, it's simple to achieve: just <code>flush</code> the serial before sleeping. This ensures that any pending serial data is sent out through the serial port before you go to sleep:</p>

<pre><code>void loop() {
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}
</code></pre>

<p>If that doesn't help you may need to turn serial off and on yourself either side of your powerdown:</p>

<pre><code>void loop() {
  Serial.begin(115200);
  Serial.println("Turning LED on");
  digitalWrite(LED_BUILTIN, HIGH);   // turn the LED on (HIGH is the voltage level)
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                      // wait for a second
  Serial.begin(115200);
  Serial.println("Turning LED off");
  digitalWrite(LED_BUILTIN, LOW);    // turn the LED off by making the voltage LOW
  Serial.flush();
  Serial.end();
  LowPower.powerDown(SLEEP_8S, ADC_OFF, BOD_OFF);                  // wait for a second
}
</code></pre>

<hr>

<p>Update: </p>

<p>I just noticed you're using the Micro. That makes a big difference. The Micro and the Leonardo's <code>Serial</code> object is actually a USB CDC/ACM serial emulation, not UART.  You cannot sustain a USB connection while sleeping. So it will be killed.</p>

<p>To use communication while sleeping you will have to switch to a proper UART connection with a USB to TTL UART (e.g., FT232RL) adaptor to connect to the TX/RX pins and use <code>Serial1</code> instead.</p>
